[User]
SM = 11. Point V is the reflection of point M across point S. Let E be the rotation of point V by angle pi/2 around point S. Let I be the rotation of point E by angle pi/4 around point S. Let J be the reflection of point I across point V. Let W be the vector from point M to point V. Let X be the perpendicular bisector of segment VJ. Point Z is the translation of point E by vector W. Point K is obtained by rotating point S by angle 5pi/12 about point V. Point R is the reflection of point E across point Z. Point F is the reflection of point M across line X. Let P be the reflection of point F across point M. Let G be the reflection of point P across point K. Point D is the reflection of point R across segment IE. Find the length of segment DG. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let UP = 2. Let M be the midpoint of segment UP. Let B be the rotation of point M by angle 2pi/3 around point U. Point Y is the midpoint of segment MB. Point Q is the reflection of point B across point M. Point X is the reflection of point U across segment YM. Find the distance between points Q and X. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let AY = 2. Let W be the reflection of point Y across point A. Let M be the vector from point A to point W. Point O is obtained by rotating point A by angle 5pi/6 about point Y. Let K be the rotation of point A by angle 5pi/6 around point O. Let N be the reflection of point K across point A. Let R be the rotation of point K by angle pi/4 around point Y. Let I be point N translated by vector M. Let X be the reflection of point W across point R. Let E be the perpendicular bisector of segment NW. Line T is the reflection of line E across point W. Let P be the line through points X and I. Let Z be the line through point R perpendicular to segment NW. Let F be the reflection of line Z across line T. Lines F and P intersect at point H. What is the length of segment WH? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let MB = 5. Let U be the rotation of point B by angle pi/12 around point M. Line T passes through points B and M. Let G be the angle bisector of angle BMU. Point I is the midpoint of segment MU. Line X passes through point B parallel to line G. Line H passes through point I parallel to line T. Point Q is the intersection of lines H and X. Let Y be the midpoint of segment QM. Find the distance between points Q and Y. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let RU = 10. Let V be the rotation of point R by angle pi/3 around point U. Let M be the midpoint of segment UR. Let N be the rotation of point R by angle 11pi/12 around point V. Point C is the reflection of point V across point R. Point Y is the reflection of point U across point N. Let T be the angle bisector of angle MYN. Let K be the line through point V perpendicular to line T. Point J is the midpoint of segment MV. Point O is the reflection of point J across line K. What is the length of segment OC? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let PX = 9. Let W be the reflection of point P across point X. Let A be the perpendicular bisector of segment PW. Let K be the reflection of line A through point W. Point L is the reflection of point X across line K. Let F be the perpendicular bisector of segment XW. Let Y be the rotation of point X by angle 2pi/3 around point P. Line B passes through point L perpendicular to segment XW. Point M is the reflection of point Y across line F. Point D is the reflection of point M across line B. Let N be the midpoint of segment DY. Find the distance between points N and M. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let NV = 9. Let O be the reflection of point N across point V. Let H be the perpendicular bisector of segment VO. Point S is the midpoint of segment NV. Line U is the reflection of line H across point S. Point C is the reflection of point O across line U. Find the length of segment OC. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
UC = 1. Point F is obtained by rotating point U by angle 11pi/12 about point C. Point Q is the reflection of point C across point F. Let V be the rotation of point F by angle pi/4 around point C. Point B is the reflection of point U across point V. Let Z be the line through points U and C. Point P is the reflection of point Q across line Z. Point Y is the midpoint of segment FP. Find the length of segment BY. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let DS = 10. Point P is the midpoint of segment DS. Let X be the rotation of point S by angle 7pi/12 around point P. Point Z is the reflection of point X across point D. Point L is the reflection of point D across point Z. Point B is the midpoint of segment PZ. Line F is the angle bisector of angle SPL. Point T is the reflection of point S across line F. What is the length of segment BT? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let VG = 5. Line P is the perpendicular bisector of segment GV. Line X passes through point G perpendicular to segment VG. Let Y be the reflection of line P across line X. Let D be the reflection of line X across line Y. Point A is on line X. Point Q is the reflection of point A across line D. What is the distance between points Q and A? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let KX = 7. Let F be the reflection of point X across point K. Point S is the midpoint of segment FK. Let O be the vector from point F to point K. Line A is the angle bisector of angle FSK. Let E be the perpendicular bisector of segment SX. Point C is on line E. Point M is the reflection of point C across line A. Let W be point C translated by vector O. Let Y be the midpoint of segment KW. Let I be the midpoint of segment MF. Point B is the reflection of point Y across line A. Find the length of segment IB. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
JL = 11. Let Y be the midpoint of segment JL. Point N is obtained by rotating point J by angle pi/6 about point L. Line W passes through points N and L. Point S is obtained by rotating point Y by angle pi/6 about point J. Let Z be the line through point S parallel to segment JL. Line C is the reflection of line Z across point N. Point R is the reflection of point Y across line C. Point X is the reflection of point R across segment JL. Let F be the perpendicular bisector of segment XL. Let K be the line through point J perpendicular to segment JL. Line V is the reflection of line K across line F. Lines V and W intersect at point M. Point E is the midpoint of segment RJ. What is the length of segment EM? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
UF = 10. Let K be the rotation of point F by angle 3pi/4 around point U. Let N be the line through point K perpendicular to segment UK. Line I passes through point F perpendicular to line N. Point R is the reflection of point K across line I. Line A is the reflection of line I across point K. Point J is the reflection of point F across segment UK. Let D be the midpoint of segment UK. Point T is the reflection of point R across line A. Let W be the midpoint of segment DJ. Point C is the reflection of point T across point W. Find the distance between points C and W. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
KM = 2. Let Z be the rotation of point K by angle pi/6 around point M. Line N passes through points K and Z. Let Y be the line through point Z perpendicular to segment MK. Let Q be the reflection of line Y across line N. Point T is the reflection of point M across line Q. Point R is the reflection of point T across segment MK. Let L be the midpoint of segment ZT. Let S be the reflection of point L across point T. Point D is the reflection of point S across segment MK. Find the length of segment RD. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let QM = 6. Let X be the perpendicular bisector of segment QM. Let G be the reflection of line X through point M. Point F lies on line G at distance 6 from point Q. Point L is the midpoint of segment FQ. Let U be the perpendicular bisector of segment LQ. Point C is the reflection of point F across line X. Point W is the reflection of point F across line U. Let N be the midpoint of segment CW. Find the length of segment NC. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let PV = 8. Point C is the midpoint of segment VP. Let L be the angle bisector of angle VCP. Let Z be the perpendicular bisector of segment CP. Let W be the reflection of point V across point P. Line A is the reflection of line Z across point P. Let H be the reflection of line L through point P. Point F is the intersection of line H and segment PW. Point N is the intersection of line A and segment PW. Find the distance between points N and F. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let IE = 5. A is constructed so that EA = 5. Let O be the reflection of point A across point E. Point L is obtained by rotating point A by angle pi/6 about point O. Let J be the vector from point A to point O. Let Z be the rotation of point O by angle pi/4 around point A. Point R is the translation of point E by vector J. Point T is obtained by rotating point Z by angle pi/4 about point L. Let C be the midpoint of segment AZ. Point W is the reflection of point Z across point R. Point Y is the midpoint of segment WT. What is the distance between points Y and C? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let MX = 6. Line R is the perpendicular bisector of segment XM. Let F be the rotation of point X by angle 5pi/6 around point M. Line V passes through points X and F. Let I be the midpoint of segment FX. Point S is obtained by rotating point I by angle 5pi/6 about point X. Let D be the reflection of line V across line R. Let Z be the line through point S parallel to line D. Let B be the reflection of point X across point S. Let C be the rotation of point B by angle 5pi/6 around point M. Let E be the rotation of point C by angle pi/6 around point M. Let A be the reflection of point C across point B. Point H is the midpoint of segment EA. Point W is the reflection of point F across line Z. Point J is the reflection of point W across point I. What is the length of segment HJ? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let IF = 3. Point O is the midpoint of segment IF. Point V is obtained by rotating point I by angle 11pi/12 about point O. Let B be the perpendicular bisector of segment FO. Let K be the rotation of point V by angle 11pi/12 around point O. Line A is the perpendicular bisector of segment FI. Line U passes through point K parallel to line A. Let N be the reflection of line U through point V. Point T is on line N. Point S lies on line B. Point H is the reflection of point O across segment TS. Point Z is the reflection of point V across segment TS. Find the length of segment HZ. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let BJ = 6. Point L is the reflection of point J across point B. Let Z be the vector from point B to point J. Point W is obtained by rotating point L by angle 5pi/12 about point B. Let Q be the rotation of point W by angle pi/2 around point L. Point O is the translation of point Q by vector Z. Point F is the midpoint of segment OW. Point G is the translation of point J by vector Z. Let A be the midpoint of segment GF. Line P is the perpendicular bisector of segment FO. Point N is the midpoint of segment QW. Line K is the reflection of line P across point N. Point M is the reflection of point A across line K. What is the distance between points M and N? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
XS = 1. Line C passes through points S and X. Let O be the line through point X perpendicular to line C. Let U be the rotation of point X by angle pi/3 around point S. Point F is the reflection of point S across line O. Let E be the reflection of point S across point U. Point M is the reflection of point F across segment ES. Find the length of segment EM. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
RL = 4. Point C is the midpoint of segment RL. Let W be the vector from point R to point L. Point M is obtained by rotating point L by angle 3pi/4 about point R. Let X be the angle bisector of angle MRC. Point T is the midpoint of segment LC. Line P passes through point T perpendicular to line X. Let I be point M translated by vector W. Point G is the reflection of point C across line P. Let J be the rotation of point R by angle pi/2 around point L. Let S be the line through point G perpendicular to segment GI. Point Y is the reflection of point R across line S. Let U be the line through point I perpendicular to segment GI. Point B is the reflection of point J across line U. Find the distance between points B and Y. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let BC = 8. Point U is obtained by rotating point C by angle 11pi/12 about point B. Line G is the angle bisector of angle BCU. Point P is the reflection of point U across line G. Let F be the line through point P parallel to line G. Line Z passes through point U parallel to line F. Point W is the reflection of point P across line Z. Find the distance between points W and B. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let WR = 4. Point X is obtained by rotating point W by angle pi/4 about point R. Point N is obtained by rotating point R by angle pi/12 about point X. Line B passes through points N and X. Let J be the midpoint of segment NW. Point Z is the reflection of point J across line B. Let T be the reflection of point X across point Z. Find the distance between points T and W. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
LP = 11. Let O be the midpoint of segment LP. Line R is the perpendicular bisector of segment PL. Point G is obtained by rotating point L by angle 11pi/12 about point O. Let F be the line through point G perpendicular to line R. Line T is the perpendicular bisector of segment GL. Let V be the reflection of line T across line F. Point N is the reflection of point G across line V. What is the distance between points N and O? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
UJ = 4. Let V be the vector from point J to point U. Let A be the reflection of point J across point U. Point N is obtained by rotating point U by angle 5pi/12 about point J. Let B be the rotation of point N by angle 7pi/12 around point U. Let X be point A translated by vector V. Let W be the rotation of point X by angle 5pi/12 around point B. Let L be the midpoint of segment NW. Let F be the midpoint of segment JW. Let M be the midpoint of segment WF. Find the measure of angle FLM, in radians. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
AT = 2. Let C be the reflection of point A across point T. Point F is the reflection of point T across point A. Line H is the perpendicular bisector of segment TC. Let P be the rotation of point A by angle 7pi/12 around point T. Point W is the reflection of point A across point F. Let Q be the perpendicular bisector of segment WT. Line E is the reflection of line Q across line H. Point S is the reflection of point P across line E. What is the distance between points S and C? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
RE = 12. Point H is obtained by rotating point R by angle 5pi/6 about point E. Point M is obtained by rotating point E by angle pi/12 about point H. Point W is the reflection of point E across point M. Point G is the reflection of point H across segment MW. Point O is the reflection of point G across point W. Find the distance between points G and O. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
JW = 3. Line H passes through point J perpendicular to segment JW. Point U is on line H with |UJ| = 7. Point D is the reflection of point J across point U. Line B passes through point U perpendicular to line H. Point P is the reflection of point W across line B. Line E is the perpendicular bisector of segment JP. Point C is the reflection of point U across line E. Find the distance between points C and D. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
XP = 10. Point E is obtained by rotating point X by angle 5pi/12 about point P. Point T is the reflection of point X across point E. Let B be the perpendicular bisector of segment XT. Let V be the line through point T parallel to line B. Let G be the rotation of point P by angle 7pi/12 around point T. Line A passes through points E and G. Let R be the line through point T parallel to line A. Point I is obtained by rotating point E by angle 2pi/3 about point P. Point Q is the midpoint of segment TG. Point K lies on line A at distance 4 from point X. Point H is the midpoint of segment IG. Point L is obtained by rotating point X by angle pi/3 about point H. Line M is the reflection of line R across line V. Line Z is the reflection of line A across point L. Line N is the reflection of line Z across point K. Point O is the intersection of lines M and N. Point J is the midpoint of segment XQ. Let C be the reflection of point O across point J. What is the length of segment HC? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
CD = 8. Point I is the midpoint of segment CD. Let S be the perpendicular bisector of segment ID. Point E is the reflection of point C across line S. Line N passes through point E parallel to line S. Let V be the line through point I parallel to line N. Line H passes through points E and C. Line U is the reflection of line S across point I. Let A be the rotation of point C by angle pi/4 around point D. Point X is the reflection of point A across line V. Line W passes through point X perpendicular to line U. Point G is the intersection of lines H and U. Point O is the reflection of point G across line W. Find the distance between points C and O. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
DH = 12. Point N is obtained by rotating point H by angle pi/2 about point D. Point V is obtained by rotating point N by angle 7pi/12 about point H. Point K is the reflection of point D across point H. Point B is obtained by rotating point N by angle pi/2 about point H. Let T be the angle bisector of angle VKH. Line Y passes through point B parallel to line T. Line U passes through points V and D. Point R is the midpoint of segment DH. Let L be the line through point R parallel to line U. Line P passes through point H parallel to line L. Let Z be the reflection of point R across point K. Point I is the intersection of lines Y and P. Point M is the reflection of point Z across point I. Let A be the reflection of point V across point K. What is the length of segment MA? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
WK = 6. Let A be the reflection of point W across point K. Let D be the midpoint of segment WK. Point X is obtained by rotating point A by angle 7pi/12 about point W. Point I is the midpoint of segment KX. Let R be the midpoint of segment KD. Let F be the reflection of point X across point A. Point S is the midpoint of segment IR. Point N is the reflection of point F across point S. What is the length of segment NI? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let FX = 10. Line E is the perpendicular bisector of segment FX. Let W be the midpoint of segment FX. Point Z is on line E with |ZX| = 12. Line A passes through point Z perpendicular to line E. Point D lies on segment FX. Line J is the reflection of line A across point D. Point Y is the reflection of point F across line J. Find the distance between points W and Y. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
RL = 2. Let W be the midpoint of segment LR. Let S be the rotation of point L by angle pi/2 around point W. Let F be the midpoint of segment LS. Point O is the midpoint of segment FL. Let X be the midpoint of segment WL. Point N is the midpoint of segment XO. Find the length of segment NS. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let CS = 9. Point I is obtained by rotating point C by angle 11pi/12 about point S. Line G is the angle bisector of angle ISC. Point K is the midpoint of segment SC. Line Z is the angle bisector of angle IKC. Let E be the rotation of point I by angle 7pi/12 around point K. Let P be the perpendicular bisector of segment ES. Line B is the reflection of line P across line G. Let W be the reflection of line G across line Z. Line Y is the reflection of line B across point K. Point F is the midpoint of segment EC. Lines Y and W intersect at point O. Find the length of segment OF. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let VY = 4. Point T is the reflection of point V across point Y. Let K be the rotation of point Y by angle 11pi/12 around point T. Let C be the line through points V and K. Point L is the reflection of point K across point Y. Let E be the perpendicular bisector of segment LY. Point I is the midpoint of segment LY. Let G be the reflection of line E through point T. Point W is the reflection of point K across point V. Point D is the midpoint of segment TK. Let A be the midpoint of segment IV. Point P is the intersection of lines C and G. Point M is the reflection of point A across segment DW. What is the length of segment MP? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ZK = 4. Let I be the rotation of point K by angle 5pi/12 around point Z. Let M be the perpendicular bisector of segment KI. Let D be the line through point K parallel to line M. Point F is the reflection of point I across line D. Point L is the midpoint of segment FZ. Let O be the midpoint of segment IL. Let G be the rotation of point K by angle 7pi/12 around point F. Point B is obtained by rotating point F by angle pi/6 about point L. Let V be the vector from point I to point G. Point U is the translation of point B by vector V. Let J be the midpoint of segment LF. Point H is the midpoint of segment UJ. Line Y is the angle bisector of angle OHK. Let W be the line through point I parallel to line Y. Line W and segment GH intersect at point Q. What is the distance between points I and Q? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let JL = 10. Let G be the rotation of point L by angle pi/3 around point J. Point Y is the reflection of point G across point J. Point U is obtained by rotating point G by angle pi/3 about point Y. Point K is obtained by rotating point Y by angle pi/2 about point J. Point I is the midpoint of segment KU. Point Q is the reflection of point J across segment LI. Point T is the reflection of point K across segment LI. Let O be the midpoint of segment QT. Point N is the midpoint of segment TO. Point S is the midpoint of segment LI. Find the measure of angle SON, in radians. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
QS = 10. Let Y be the rotation of point Q by angle pi/3 around point S. Let T be the midpoint of segment SY. Let M be the midpoint of segment YQ. Let U be the midpoint of segment TQ. Point O is the reflection of point S across segment UY. Find the distance between points M and O. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
LF = 9. Point E is the reflection of point F across point L. Let J be the midpoint of segment FL. Let Y be the midpoint of segment EJ. Line U is the angle bisector of angle LYE. Point X is the reflection of point L across line U. Point M is on line U with |ML| = 5. Point I is the reflection of point Y across point E. Point S is the midpoint of segment XM. Let V be the reflection of point I across point F. Find the distance between points S and V. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
DH = 5. Let I be the rotation of point D by angle 2pi/3 around point H. Point U is obtained by rotating point D by angle 5pi/12 about point H. Point P is obtained by rotating point I by angle 5pi/6 about point D. Point K is obtained by rotating point I by angle pi/3 about point P. Line W passes through point D parallel to segment UP. Point A is obtained by rotating point I by angle 5pi/12 about point K. Line N is the perpendicular bisector of segment PA. Let T be the reflection of line N through point I. Point X is the intersection of lines W and T. Point O is the midpoint of segment UP. Find the distance between points O and X. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let HM = 9. Point T is the reflection of point H across point M. Point E is obtained by rotating point M by angle pi/6 about point H. Point S is the midpoint of segment HE. Point V is the reflection of point S across point T. Let C be the perpendicular bisector of segment VE. Let O be the reflection of point T across point H. Point Y is the reflection of point H across line C. Let R be the midpoint of segment MO. Point D is the midpoint of segment HR. What is the length of segment YD? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let EM = 5. Let J be the rotation of point E by angle 7pi/12 around point M. Let D be the rotation of point J by angle 5pi/6 around point E. Let S be the midpoint of segment MJ. Let F be the rotation of point S by angle pi/3 around point J. Let C be the reflection of point D across point F. Let X be the reflection of point C across point D. Point R is the midpoint of segment SD. Point O is the midpoint of segment MR. Find the length of segment XO. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
TR = 2. Point C is the reflection of point R across point T. Point K is the reflection of point T across point C. Point Z is obtained by rotating point T by angle pi about point K. Let Q be the rotation of point K by angle 2pi/3 around point R. Line H is the angle bisector of angle QTZ. Let M be the reflection of line H through point C. Point G lies on line M at distance 11 from point Z. Let W be the midpoint of segment RG. Point A is the reflection of point W across line M. Point Y is the midpoint of segment GC. Point N is the reflection of point A across segment GC. Find the distance between points N and Y. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let UG = 2. Point C is the midpoint of segment UG. Let B be the midpoint of segment CG. Let O be the midpoint of segment BG. Let J be the midpoint of segment OB. Find the distance between points O and J. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let NP = 8. Point S is the midpoint of segment PN. Let Q be the midpoint of segment SP. Let R be the reflection of point Q across point S. Let I be the angle bisector of angle PQR. Point O is obtained by rotating point S by angle 5pi/12 about point Q. Point H is the midpoint of segment RN. Point C is the reflection of point O across line I. Line W is the angle bisector of angle OPC. Line K is the reflection of line W across point H. Point F is the reflection of point Q across line K. Find the length of segment FH. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let LY = 1. Point E is the midpoint of segment YL. Point J is obtained by rotating point L by angle 11pi/12 about point E. Point D is obtained by rotating point Y by angle 5pi/6 about point J. Let Q be the rotation of point L by angle pi/3 around point Y. Let X be the rotation of point Q by angle 5pi/12 around point D. Point A is the reflection of point J across point Q. Point N is the midpoint of segment EX. Point K is the midpoint of segment QN. What is the distance between points K and A? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let GN = 5. Point M is obtained by rotating point N by angle 7pi/12 about point G. Line H passes through point M perpendicular to segment NM. Let W be the line through point G perpendicular to line H. Point V is the reflection of point N across line W. Let U be the reflection of point G across point N. Let L be the rotation of point V by angle 3pi/4 around point U. Let S be the perpendicular bisector of segment NU. Point J is the reflection of point L across line S. Point Q is the midpoint of segment UJ. Find the length of segment GQ. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
WF = 2. Point M is obtained by rotating point W by angle pi/4 about point F. Point K is obtained by rotating point W by angle 5pi/12 about point F. Point C is the reflection of point K across point W. Let D be the midpoint of segment KW. Line P is the angle bisector of angle DMF. Line N is the reflection of line P across point K. Point Z is the reflection of point C across segment MF. Point U is the reflection of point K across line N. Find the distance between points Z and U. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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Let DF = 8. Let G be the perpendicular bisector of segment DF. Let Z be the line through points F and D. Let R be the line through point D parallel to line G. Line I passes through point F parallel to line R. Point S lies on line I at distance 5 from point F. Let Q be the rotation of point S by angle pi around point D. Let O be the midpoint of segment DS. Point K is the reflection of point Q across line Z. Let V be the reflection of point Q across point S. Point Y is the midpoint of segment KO. Find the distance between points Y and V. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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Let IM = 5. Line L is the perpendicular bisector of segment MI. Let Z be the line through point M parallel to line L. Let A be the reflection of line L across line Z. Point E is obtained by rotating point M by angle pi/12 about point I. Let J be the reflection of point E across point I. Let H be the line through point E perpendicular to line Z. Let D be the line through points M and E. Point V is the reflection of point I across point J. Lines D and A intersect at point T. Line N is the perpendicular bisector of segment TE. Point Y is the reflection of point M across line H. Point G is the midpoint of segment YT. Point Q is the reflection of point G across line N. Find the length of segment QV. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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LR = 3. Let N be the reflection of point R across point L. Let I be the rotation of point N by angle 5pi/12 around point L. Let B be the rotation of point R by angle 7pi/12 around point N. Point W is obtained by rotating point B by angle pi/2 about point N. Point Z is the midpoint of segment NB. Point V is the reflection of point Z across segment LB. Point K is the reflection of point W across segment LB. Let U be the midpoint of segment KI. What is the distance between points V and U? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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WU = 12. Let L be the vector from point W to point U. Let Z be point U translated by vector L. Point O is obtained by rotating point Z by angle 3pi/4 about point W. Point G is obtained by rotating point Z by angle 5pi/12 about point U. Let C be the perpendicular bisector of segment ZO. Point F is obtained by rotating point Z by angle 5pi/6 about point O. Point E is the reflection of point U across point W. Line V passes through point F parallel to line C. Point N is the reflection of point G across point Z. Point B is the reflection of point G across line V. Point M is the reflection of point N across segment OZ. Let R be the reflection of point E across point G. Point Q is the reflection of point R across segment MB. Point K is the midpoint of segment MB. Find the length of segment KQ. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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Let PR = 10. Line X passes through points R and P. Point O lies on line X at distance 5 from point P. Point D is obtained by rotating point O by angle pi/6 about point P. Let A be the vector from point D to point P. Point I is the translation of point P by vector A. Let G be the perpendicular bisector of segment OP. Point B is the reflection of point O across point P. Let V be the line through points I and B. Let S be the reflection of line G through point D. Point E is the midpoint of segment PI. Lines S and V intersect at point U. What is the length of segment UE? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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HE = 1. Let U be the line through points H and E. Let Z be the rotation of point E by angle pi/12 around point H. Point X is the midpoint of segment ZE. Line K passes through point Z parallel to line U. Line R passes through points H and Z. Point Y is the reflection of point X across line R. Let D be the reflection of line U across line K. Point N is the reflection of point Z across line D. Let M be the reflection of point Y across point N. Point T is the reflection of point E across line K. Point S is the reflection of point M across segment XH. What is the distance between points S and T? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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MQ = 2. Let P be the perpendicular bisector of segment MQ. Let A be the midpoint of segment QM. Point G is the reflection of point A across point Q. Point Y lies on line P at distance 11 from point G. Line R passes through point Y parallel to segment MQ. Line B is the reflection of line R across point Q. Point K is the reflection of point Q across line B. Point N is the reflection of point M across line B. Point V is the reflection of point K across point Q. Point E is the midpoint of segment NY. What is the length of segment VE? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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QL = 3. Point N is obtained by rotating point L by angle 3pi/4 about point Q. Point Z is the reflection of point Q across point L. Point W is the midpoint of segment NZ. Let H be the midpoint of segment QW. What is the distance between points H and Z? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
JT = 3. Line Q passes through points T and J. Let F be the rotation of point T by angle pi/12 around point J. Point V is obtained by rotating point J by angle pi/3 about point F. Line O is the perpendicular bisector of segment FV. Line N passes through point F parallel to line O. Let L be the reflection of point F across point J. Line H passes through point L parallel to line O. Point S is the reflection of point T across line N. Point I is the intersection of lines H and Q. Find the length of segment IS. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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Let AF = 12. Line E passes through points A and F. Let K be the rotation of point F by angle pi/6 around point A. Point O is obtained by rotating point A by angle 3pi/4 about point F. Let R be the reflection of point K across point O. Let B be the reflection of point R across point A. Point I is the reflection of point K across line E. Point T is the midpoint of segment IO. Point J is the midpoint of segment AB. Let Z be the midpoint of segment TJ. Find the length of segment BZ. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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FJ = 10. Point A is the midpoint of segment JF. Let B be the rotation of point A by angle pi/12 around point F. Line I is the perpendicular bisector of segment AF. Line V passes through point B perpendicular to line I. Let T be the line through points J and B. Point K lies on segment JF. Point M is the reflection of point K across line T. Point Q is the reflection of point K across line V. Point D is the reflection of point Q across line T. What is the length of segment MD? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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IV = 6. Let Z be the midpoint of segment IV. Let L be the midpoint of segment ZI. Point A is obtained by rotating point I by angle pi/12 about point L. Let J be the reflection of point V across point L. Let G be the midpoint of segment AJ. Point R is the reflection of point G across point J. What is the distance between points R and Z? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let KC = 10. Point V is obtained by rotating point C by angle 5pi/6 about point K. Let T be the rotation of point V by angle 3pi/4 around point K. Point E is the midpoint of segment TK. Let Q be the line through points E and V. Line B is the angle bisector of angle TVC. Let L be the line through point C parallel to line B. Point Y is the intersection of lines Q and L. Let H be the reflection of point T across point K. Find the distance between points Y and H. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ET = 9. Let Q be the midpoint of segment ET. Let F be the perpendicular bisector of segment TQ. Line W passes through point T parallel to line F. Point X is the midpoint of segment QE. Point H is the reflection of point E across line W. Let M be the rotation of point E by angle 5pi/12 around point T. Point P is the midpoint of segment XM. Point J is obtained by rotating point H by angle 5pi/6 about point Q. Point G is the reflection of point T across point E. Let Y be the line through point T parallel to segment PQ. Point N is the reflection of point J across line Y. Let L be the midpoint of segment GN. What is the length of segment LH? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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Let XT = 4. Point E is obtained by rotating point T by angle pi/4 about point X. Point M is the reflection of point X across point E. Let W be the reflection of point E across point X. Point Q is the reflection of point W across segment TE. Point F is the midpoint of segment TE. Let R be the reflection of point M across point Q. What is the length of segment RF? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let UE = 5. Point Z is obtained by rotating point U by angle pi/6 about point E. Point K is the reflection of point U across point Z. Line D is the angle bisector of angle EZU. Point O is obtained by rotating point U by angle 11pi/12 about point K. Let S be the midpoint of segment EK. Line P is the reflection of line D across point S. Let A be the rotation of point S by angle pi/6 around point O. Line Y is the reflection of line P across line D. Line N is the perpendicular bisector of segment AS. Let Q be the reflection of line N through point S. Point F is the intersection of lines Y and Q. What is the length of segment UF? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
QX = 1. Let N be the reflection of point X across point Q. Let M be the perpendicular bisector of segment XQ. Point Z is the reflection of point N across line M. Let A be the line through points X and Z. Let V be the line through point Q perpendicular to line A. Point J is the reflection of point Z across line V. Find the distance between points J and N. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let HX = 9. Let Q be the vector from point H to point X. Let J be the perpendicular bisector of segment XH. Let O be the rotation of point X by angle pi around point H. Let E be point X translated by vector Q. Point U lies on line J at distance 8 from point E. Line Z passes through points U and E. Point N is the reflection of point E across point O. Let D be the rotation of point U by angle pi around point N. Point F is the reflection of point X across point E. Line P passes through point D parallel to segment XH. Let C be the reflection of line Z across line P. Point B is the reflection of point E across line C. Find the length of segment FB. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let YL = 9. Let X be the vector from point Y to point L. Let B be the midpoint of segment LY. Let E be point B translated by vector X. Let C be the rotation of point E by angle 3pi/4 around point L. Let J be point C translated by vector X. Point N is the reflection of point C across point B. Let M be the midpoint of segment JB. Let V be the reflection of point M across point C. Point R is the reflection of point Y across segment NL. Find the distance between points R and V. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let SF = 6. Point C is the midpoint of segment FS. Let B be the perpendicular bisector of segment FS. Let G be the line through point F parallel to line B. Let P be the rotation of point F by angle 5pi/12 around point S. Let T be the midpoint of segment PS. Let E be the rotation of point S by angle pi/4 around point P. Let M be the reflection of point C across point S. Let N be the reflection of point F across point P. Point Y is the reflection of point T across line G. Line I passes through point Y perpendicular to segment CE. Let L be the line through point Y perpendicular to line I. Line V is the reflection of line L across point M. Point H is the reflection of point P across line V. Point R is the reflection of point H across line L. Find the length of segment NR. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ZG = 1. Point Y is obtained by rotating point G by angle 11pi/12 about point Z. Line I is the angle bisector of angle ZGY. Let B be the perpendicular bisector of segment ZY. Point D is the midpoint of segment ZY. Line C is the reflection of line I across line B. Let V be the midpoint of segment GD. Let P be the line through point Z perpendicular to line C. Let R be the line through point V perpendicular to segment YZ. Let S be the reflection of line R across line P. Point X is the reflection of point D across line P. Point Q is the intersection of line S and segment YZ. Point O is the reflection of point G across segment YZ. Point F is the reflection of point Z across segment QG. Point K is the reflection of point O across segment VX. What is the length of segment KF? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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RX = 2. Point K is obtained by rotating point X by angle pi/3 about point R. Line O is the perpendicular bisector of segment XK. Let J be the line through points R and X. Line B passes through point R perpendicular to line O. Point A is the reflection of point K across line B. Let Z be the reflection of line O through point K. Let N be the rotation of point K by angle pi/6 around point A. Let T be the midpoint of segment XR. Point W is the reflection of point N across line Z. Let P be the perpendicular bisector of segment WT. Let Y be the reflection of line J across line P. Point Q is the reflection of point X across line Y. What is the length of segment NQ? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let KJ = 6. Point H is the reflection of point K across point J. Let R be the rotation of point H by angle 2pi/3 around point K. Point S is the reflection of point K across point R. Let P be the vector from point S to point R. Point B is the translation of point H by vector P. Let I be the rotation of point R by angle 7pi/12 around point S. Line W passes through point B parallel to segment SK. Line C passes through point J perpendicular to line W. Point Y is the reflection of point H across line C. Point T is the reflection of point I across point Y. What is the length of segment IT? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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XT = 2. Let D be the rotation of point X by angle 7pi/12 around point T. Let C be the midpoint of segment TX. Point Z is the midpoint of segment CD. Point P is the reflection of point Z across segment CT. Point M is the midpoint of segment CT. Point A is the reflection of point P across point Z. Point B is the reflection of point A across point C. What is the length of segment MB? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
HW = 11. Point K is obtained by rotating point H by angle pi/12 about point W. Let P be the midpoint of segment HK. Let A be the angle bisector of angle PWK. Point C is the reflection of point P across line A. Point Z is the reflection of point C across point P. What is the length of segment CZ? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let BZ = 4. Let G be the vector from point Z to point B. Let J be the reflection of point Z across point B. Line S is the angle bisector of angle JBZ. Point Q is the translation of point J by vector G. Line E is the reflection of line S across point Q. Line F is the reflection of line E across line S. Let D be the reflection of point Q across point B. Let K be the line through point Q parallel to line F. Point L is the reflection of point D across line K. Line P passes through point Z parallel to line F. Let V be the midpoint of segment ZB. Point M is the reflection of point Q across point V. Let U be the midpoint of segment ZL. Point T is the reflection of point L across line P. Let H be the midpoint of segment UM. Find the distance between points H and T. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
OW = 6. Line N passes through point O perpendicular to segment WO. Let M be the perpendicular bisector of segment OW. Let S be the line through point W parallel to line N. Let T be the midpoint of segment WO. Point V is the reflection of point T across line S. Let Y be the rotation of point O by angle 11pi/12 around point W. Let A be the rotation of point T by angle 3pi/4 around point O. Let B be the perpendicular bisector of segment VW. Line Z is the reflection of line B across line M. Point K is the intersection of line Z and segment WA. Let X be the midpoint of segment KY. Let I be the reflection of point Y across point K. What is the distance between points X and I? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
IF = 1. Let X be the rotation of point I by angle 5pi/6 around point F. Let Z be the reflection of point I across point X. Let Y be the midpoint of segment FX. Line H is the perpendicular bisector of segment ZY. Point E is the reflection of point F across line H. Point M is the midpoint of segment EY. Find the length of segment EM. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let FI = 10. Let J be the midpoint of segment FI. Let S be the vector from point I to point F. Point Z is the midpoint of segment IJ. Line L passes through point Z perpendicular to segment JF. Line G passes through point J parallel to line L. Let K be the reflection of line L through point F. Point B is the translation of point Z by vector S. Point E is on line K with |EF| = 12. Point M is the midpoint of segment IB. Line Q is the reflection of line K across point J. Line D is the reflection of line G across line Q. Point P is the reflection of point E across line D. What is the length of segment PM? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
AB = 7. Let F be the reflection of point A across point B. Let K be the midpoint of segment BA. Line J passes through point F perpendicular to segment BA. Let P be the reflection of line J through point A. Point G is the reflection of point K across line P. Line I passes through point G parallel to line J. Point M is obtained by rotating point K by angle pi/6 about point G. Point N is the midpoint of segment AM. Point Z is obtained by rotating point F by angle 7pi/12 about point K. Point R is the reflection of point Z across line I. Point X is the reflection of point N across line J. Line Y passes through points X and N. Point S is the reflection of point B across line Y. Let Q be the midpoint of segment SX. What is the length of segment RQ? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
VF = 11. Let H be the midpoint of segment VF. Line U is the angle bisector of angle FHV. Point Z lies on line U at distance 8 from point H. Let C be the line through points H and F. Line B is the perpendicular bisector of segment FZ. Line G is the perpendicular bisector of segment HF. Point T is the reflection of point Z across point V. Point A is the reflection of point V across line B. Point E is the reflection of point T across point Z. Point W is the reflection of point F across point E. Point P is the reflection of point A across segment WH. Line K passes through point P parallel to line G. Point I is the intersection of lines K and C. Let X be the line through point W parallel to line K. Point S is the intersection of lines C and X. Find the distance between points I and S. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let YS = 11. Point U is the midpoint of segment YS. Point O is obtained by rotating point Y by angle pi/4 about point U. Let Z be the rotation of point S by angle 2pi/3 around point O. Point W is the reflection of point Y across point O. Point N is the reflection of point U across point W. Point C is the reflection of point O across segment UW. Point J is the midpoint of segment NZ. Point F is the midpoint of segment ZC. What is the length of segment FJ? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
QS = 12. Point I is obtained by rotating point Q by angle pi/12 about point S. Let L be the vector from point S to point I. Point T is the translation of point I by vector L. Point E lies on segment SQ. Point N is obtained by rotating point I by angle 3pi/4 about point Q. Let C be the midpoint of segment ET. Let M be the line through point T perpendicular to segment TI. Point A is the reflection of point Q across line M. Point G is the reflection of point N across point A. Point X is the midpoint of segment EG. Point Z is the reflection of point X across segment SQ. Point R is the midpoint of segment CZ. Find the distance between points Z and R. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
AF = 1. Point N is the midpoint of segment FA. Point V is obtained by rotating point A by angle pi about point F. Let S be the angle bisector of angle FNA. Let D be the line through point F parallel to line S. Point C is the midpoint of segment VF. Let Y be the rotation of point A by angle 2pi/3 around point C. Line K is the reflection of line D across line S. Let E be the reflection of point N across point C. Point Z is the reflection of point C across line K. Let L be the midpoint of segment YE. Find the measure of angle ZVL, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let BG = 5. Point K is obtained by rotating point B by angle pi/4 about point G. Point X is the midpoint of segment KG. Line R passes through point G perpendicular to segment BG. Point N is the reflection of point K across line R. Let L be the reflection of point X across point N. Find the length of segment XL. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
BC = 1. Let Q be the midpoint of segment CB. Line E passes through point C perpendicular to segment CB. Point P is on line E with |PC| = 10. Point M is the midpoint of segment QB. Point D is the reflection of point P across point M. Let I be the line through point D parallel to line E. Point J is the reflection of point B across line I. What is the distance between points M and J? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let WI = 8. Point X is obtained by rotating point W by angle 11pi/12 about point I. Line H passes through points W and X. Line S is the perpendicular bisector of segment IX. Point N is the reflection of point I across line H. Let B be the vector from point N to point I. Point E is the reflection of point W across line S. Line O is the perpendicular bisector of segment IW. Let U be the perpendicular bisector of segment NX. Let C be point I translated by vector B. Let Y be the rotation of point C by angle 11pi/12 around point N. Point A is the midpoint of segment NE. Line R passes through point C parallel to line U. Segment NE and line R intersect at point F. Point D is the midpoint of segment XN. Point J is the reflection of point Y across line O. Point P is the reflection of point I across point N. Line G is the perpendicular bisector of segment AF. Point T is the reflection of point P across segment NE. Point K is the reflection of point J across line G. Let L be the midpoint of segment DT. What is the distance between points K and L? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
GL = 9. Line F passes through points G and L. Point J lies on line F at distance 10 from point G. Let S be the midpoint of segment JG. Line M passes through point S perpendicular to line F. Line O is the perpendicular bisector of segment SL. Point W lies on line O at distance 11 from point L. Point B is on line M with |BL| = 8. Point V is the midpoint of segment WS. Point X is the reflection of point G across point V. Let Y be the midpoint of segment JB. Let N be the angle bisector of angle XGY. Point A is the reflection of point Y across line N. Point Q is the reflection of point A across point G. Find the length of segment QA. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let TQ = 6. Point S is the reflection of point Q across point T. Let V be the rotation of point S by angle pi around point Q. Let E be the rotation of point V by angle 11pi/12 around point Q. Point P is obtained by rotating point Q by angle 5pi/12 about point E. Let Y be the midpoint of segment VT. Let N be the line through points V and P. Point B is the reflection of point Y across line N. Find the length of segment EB. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let YU = 12. Let P be the reflection of point Y across point U. Let S be the midpoint of segment UY. Let W be the reflection of point U across point Y. Line X is the angle bisector of angle PUS. Let K be the angle bisector of angle UYW. Point A is the reflection of point S across line K. Point H is the reflection of point S across line X. Find the distance between points A and H. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
FW = 1. Point V is obtained by rotating point W by angle 3pi/4 about point F. Point I is the reflection of point V across segment WF. Let X be the line through points I and W. Point J is the reflection of point V across line X. Let S be the reflection of point V across point W. Point M is the reflection of point J across point S. Let P be the reflection of point S across point V. Let G be the reflection of point M across point P. Find the distance between points V and G. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let FU = 2. Line S is the perpendicular bisector of segment UF. Let C be the midpoint of segment UF. Let L be the reflection of point C across point F. Point M is obtained by rotating point F by angle 11pi/12 about point C. Let G be the line through point F parallel to line S. Let X be the reflection of point L across point M. Let T be the vector from point U to point L. Point Y is the translation of point M by vector T. Let H be the line through points C and Y. Point J is the midpoint of segment CX. Lines H and G intersect at point R. Point Z is the midpoint of segment XY. Let Q be the reflection of point J across point C. Point N is the midpoint of segment RZ. What is the distance between points Q and N? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let YL = 5. Let J be the line through point Y perpendicular to segment LY. Point U lies on line J at distance 3 from point L. Let S be the angle bisector of angle LYU. Line N is the reflection of line S across point U. Point V is the reflection of point U across line N. Point I is the reflection of point V across segment LY. What is the distance between points V and I? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let KB = 3. Let W be the reflection of point K across point B. Point F is obtained by rotating point K by angle 2pi/3 about point W. Let D be the perpendicular bisector of segment FW. Let V be the reflection of line D through point F. Point S is the reflection of point K across line V. Let I be the reflection of point B across point F. Find the length of segment SI. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
CJ = 3. Let T be the rotation of point J by angle 7pi/12 around point C. Let P be the reflection of point T across point C. Let K be the vector from point P to point C. Point M is the translation of point J by vector K. Point I is obtained by rotating point C by angle 3pi/4 about point P. Point X is the midpoint of segment JI. Line G passes through point P parallel to segment IT. Let V be point X translated by vector K. Point Y is obtained by rotating point V by angle 11pi/12 about point I. Let D be the midpoint of segment TM. Point E is the reflection of point Y across segment DI. Point N is the reflection of point E across line G. What is the length of segment EN? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let VY = 11. Let P be the line through points V and Y. Point D is obtained by rotating point Y by angle 5pi/12 about point V. Let M be the reflection of point D across point V. Line G is the perpendicular bisector of segment VD. Line E passes through point V parallel to line G. Let Q be the line through point V perpendicular to line P. Point W is the reflection of point D across line Q. Line E and segment MW intersect at point L. Let R be the midpoint of segment MW. Find the distance between points R and L. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
QH = 3. Let P be the rotation of point H by angle 11pi/12 around point Q. Let B be the perpendicular bisector of segment QP. Point T is the midpoint of segment PQ. Let U be the line through points T and H. Point K is the reflection of point Q across point P. Let C be the reflection of line B through point H. Let Z be the reflection of point K across point T. Point I is the midpoint of segment HQ. Let O be the perpendicular bisector of segment QT. Let N be the reflection of line U across line O. Point M is the reflection of point I across line B. Line D is the reflection of line N across point M. Let W be the midpoint of segment TM. Let E be the line through point Z parallel to line C. Let L be the line through points P and W. Lines E and D intersect at point G. Point X is the reflection of point G across segment QT. Point F is the intersection of lines L and U. Find the distance between points F and X. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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QN = 2. Point B is the reflection of point N across point Q. Let A be the line through points B and Q. Let Z be the vector from point B to point Q. Point J is the midpoint of segment BQ. Let L be point J translated by vector Z. Let D be the rotation of point Q by angle 11pi/12 around point J. Let U be the rotation of point B by angle 5pi/6 around point D. Let G be the vector from point L to point U. Point F is the translation of point D by vector G. Point Y is the reflection of point U across line A. Point R is the reflection of point J across point D. Point T is the midpoint of segment FQ. Point X is the reflection of point Y across segment RT. Point O is the midpoint of segment RT. Find the distance between points X and O. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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RT = 3. Let H be the reflection of point T across point R. Let G be the midpoint of segment RT. Let L be the rotation of point T by angle pi/3 around point R. Let Z be the midpoint of segment HR. Let A be the rotation of point H by angle pi/3 around point T. Let M be the rotation of point T by angle 11pi/12 around point A. Let J be the angle bisector of angle ZMG. Let K be the line through points T and G. Line N is the reflection of line K across line J. Let F be the reflection of point G across point M. Let B be the line through point A perpendicular to line N. Point U is the reflection of point L across line B. Point Q is the reflection of point U across segment FG. Let P be the midpoint of segment UQ. What is the length of segment PR? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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HX = 2. Let O be the reflection of point X across point H. Let N be the angle bisector of angle XHO. Let G be the perpendicular bisector of segment HO. Point Z lies on line N at distance 9 from point O. Point B is the midpoint of segment HX. Let L be the reflection of line N across line G. Point U is the midpoint of segment BX. Let P be the reflection of point O across point Z. Let M be the midpoint of segment HU. Line W passes through point P parallel to segment HX. Point K is the reflection of point M across line L. Point V is the reflection of point K across point B. Lines L and W intersect at point E. What is the distance between points V and E? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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SF = 7. J is placed 6 units away from F. Line W is the perpendicular bisector of segment JS. Point Q is the reflection of point F across line W. Point D is the midpoint of segment QJ. Let U be the midpoint of segment JS. Find the distance between points D and U. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let US = 1. Line M is the perpendicular bisector of segment SU. Point X is the reflection of point S across point U. Point C is the reflection of point X across line M. Let J be the reflection of point C across point X. What is the distance between points C and J? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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HI = 4. Point A is obtained by rotating point H by angle pi/2 about point I. Let C be the rotation of point I by angle 7pi/12 around point A. Let W be the perpendicular bisector of segment IC. Line B passes through point C perpendicular to line W. Line X passes through point H parallel to line W. Point Q is the reflection of point A across line X. Let M be the midpoint of segment CA. Point N is obtained by rotating point Q by angle pi/2 about point H. Point G is the midpoint of segment NM. Let O be the line through points N and Q. Point J is the reflection of point G across line O. Point V is the reflection of point J across line B. What is the length of segment VN? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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JS = 8. Line M passes through point J perpendicular to segment JS. Let Q be the reflection of point J across point S. Point C lies on line M at distance 11 from point S. Point H is the reflection of point C across segment JS. Let O be the perpendicular bisector of segment QC. Let E be the midpoint of segment SJ. Point V is the reflection of point E across line O. Point Z is the midpoint of segment HS. Let R be the reflection of point Z across point V. What is the length of segment JR? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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Let HS = 8. Point K is obtained by rotating point H by angle 7pi/12 about point S. Point U is the reflection of point S across point K. Point X is obtained by rotating point U by angle 7pi/12 about point S. Point Y is the midpoint of segment XK. Line D passes through points K and Y. Line P is the perpendicular bisector of segment UX. Let J be the reflection of point H across point X. Line M is the reflection of line P across line D. Point B is the reflection of point J across line M. Let T be the reflection of point K across point B. Point Q is the reflection of point B across point T. What is the distance between points Q and X? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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Let VZ = 9. Let L be the rotation of point V by angle 5pi/6 around point Z. Point C is the reflection of point Z across point V. Point E is the midpoint of segment LV. Point P is the reflection of point Z across point L. Line W is the angle bisector of angle EPL. Point S is the reflection of point C across line W. Point B is the midpoint of segment PS. Point X is the reflection of point S across segment ZV. What is the distance between points B and X? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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SE = 2. Let L be the rotation of point S by angle pi/4 around point E. Let F be the rotation of point L by angle 5pi/12 around point S. Line Y passes through point S perpendicular to segment LF. Let N be the reflection of point E across point S. Point V is the midpoint of segment FN. Point G is the midpoint of segment LS. Line T passes through point N perpendicular to line Y. Let K be the perpendicular bisector of segment GN. Let D be the line through point F parallel to line K. Point J is the reflection of point S across line K. Line U passes through point J perpendicular to line T. Point H is the reflection of point V across line D. Point W is the reflection of point N across line U. Let X be the midpoint of segment HW. What is the distance between points X and N? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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Let EL = 8. J is placed 5 units away from L. Point R is obtained by rotating point J by angle 7pi/12 about point E. Let G be the reflection of point L across point J. Let M be the midpoint of segment GR. Let N be the reflection of point G across point R. Point U is the midpoint of segment LN. Let Z be the midpoint of segment UL. What is the length of segment ZM? Express your answer as a decimal number rounded to 2 decimal places.

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Let VJ = 7. Point Q is obtained by rotating point V by angle 5pi/6 about point J. Let H be the line through point J perpendicular to segment VQ. Let L be the midpoint of segment VQ. VI = 9. Let T be the rotation of point L by angle 5pi/6 around point V. Let A be the reflection of point I across point L. Point B is the reflection of point A across line H. Point O is the reflection of point A across segment VQ. Point P is the reflection of point O across point T. Find the distance between points B and P. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let OQ = 10. Point K is obtained by rotating point O by angle pi/6 about point Q. Let J be the reflection of point K across point O. Let W be the line through point K perpendicular to segment OK. Let C be the line through point O parallel to line W. Line N passes through points J and Q. Let V be the midpoint of segment OK. Point F is the intersection of lines N and C. What is the distance between points V and F? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
AX = 1. Point J is the midpoint of segment XA. Let M be the rotation of point X by angle 7pi/12 around point A. Point E is the midpoint of segment XJ. Let C be the perpendicular bisector of segment XE. Let Q be the perpendicular bisector of segment AX. Let G be the reflection of line C across line Q. Point Z is the midpoint of segment AM. Point T is the intersection of line G and segment AX. Let S be the reflection of point Z across point J. Point P is the reflection of point S across line C. Point I is the reflection of point T across point A. Find the distance between points I and P. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let TB = 3. TG = 6. Point L is the midpoint of segment TB. Let S be the perpendicular bisector of segment BG. Point Q is the midpoint of segment BL. Let E be the line through point Q parallel to line S. Point W is the reflection of point L across line E. What is the distance between points W and Q? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let MJ = 12. Let F be the midpoint of segment MJ. Let V be the rotation of point M by angle 11pi/12 around point F. Let D be the midpoint of segment FV. Let K be the line through point V parallel to segment JM. Let O be the line through points D and F. Point I is the reflection of point F across point J. Point R is the reflection of point M across line O. Let T be the reflection of line K through point I. Point E is the reflection of point I across segment JR. Line T and segment JR intersect at point L. Let S be the reflection of point E across point L. Find the length of segment RS. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let MT = 8. Let G be the perpendicular bisector of segment TM. Point U is obtained by rotating point M by angle 2pi/3 about point T. Let X be the midpoint of segment UM. Point A is on line G. Point E is obtained by rotating point X by angle pi about point M. Let S be the angle bisector of angle EXA. Let W be the line through point U parallel to line S. Point C is on line W. Point B is the midpoint of segment AC. Let N be the line through points B and E. Point H is the reflection of point M across line N. Find the distance between points E and H. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
BD = 12. Point K is the midpoint of segment BD. Let J be the rotation of point B by angle 11pi/12 around point K. Line A passes through points J and D. Line E is the reflection of line A across point B. Let V be the line through point B perpendicular to line E. Let P be the line through point K parallel to line V. Let Y be the reflection of line P across line V. Lines A and Y intersect at point O. Point W is the reflection of point K across line V. Point C is the midpoint of segment OK. Find the length of segment CW. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
TJ = 12. Let I be the rotation of point J by angle 3pi/4 around point T. Line K passes through points I and T. Let R be the reflection of point I across point J. Let V be the reflection of line K through point J. Line L is the reflection of line V across point T. Let O be the midpoint of segment TR. Point Y is the reflection of point R across line L. Point B is the reflection of point Y across line K. Point P is the reflection of point B across segment JO. Find the length of segment YP. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
BW = 10. Let Z be the midpoint of segment WB. Let M be the rotation of point W by angle 3pi/4 around point Z. Let F be the angle bisector of angle MBZ. Point G is the midpoint of segment BM. Let P be the line through point G parallel to line F. Point X is the reflection of point G across point Z. Let T be the line through point X parallel to line P. Line N is the reflection of line T across line F. Point K is the reflection of point G across line N. Find the distance between points K and B. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
TJ = 10. Line W is the perpendicular bisector of segment TJ. Point N is on line W with |NJ| = 2. Point C is the reflection of point J across point N. Point P is the midpoint of segment TC. Point S is the reflection of point P across segment TN. Point K is the intersection of segments SP and TN. Point I is the reflection of point K across point S. What is the distance between points I and S? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let EY = 2. Point Q is obtained by rotating point Y by angle pi/2 about point E. Point L is the reflection of point Q across point Y. Let G be the midpoint of segment EQ. Let I be the midpoint of segment YG. Point D is obtained by rotating point Y by angle 2pi/3 about point L. Let X be the reflection of point L across point I. Let J be the midpoint of segment GD. Point K is the reflection of point X across segment JD. What is the length of segment KX? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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Let FP = 5. Line Q is the perpendicular bisector of segment PF. Point J is on line Q with |JP| = 5. Point H is the midpoint of segment FJ. Point X is the reflection of point H across point P. Let U be the midpoint of segment FX. Let O be the midpoint of segment PX. What is the length of segment UO? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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Let KT = 11. Point E is obtained by rotating point T by angle 2pi/3 about point K. Let D be the perpendicular bisector of segment KE. Let N be the midpoint of segment TE. Point U lies on line D. Let L be the midpoint of segment UN. Point R is the reflection of point L across line D. Find the distance between points L and R. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
LK = 5. Let Q be the rotation of point K by angle 7pi/12 around point L. Let W be the rotation of point K by angle 3pi/4 around point Q. Let S be the rotation of point W by angle pi/12 around point Q. Point D is obtained by rotating point S by angle 5pi/12 about point L. Point J is the midpoint of segment LD. Let O be the midpoint of segment KD. Line I is the perpendicular bisector of segment DQ. Line B is the angle bisector of angle JKQ. Line P passes through point O perpendicular to line I. Point R is obtained by rotating point Q by angle 5pi/6 about point L. Point C is the midpoint of segment OR. Line M is the reflection of line P across line B. Line A passes through point Q perpendicular to line M. Point N is the reflection of point R across line A. What is the length of segment CN? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
OR = 6. Let E be the reflection of point O across point R. Let M be the rotation of point O by angle pi/4 around point E. Let T be the midpoint of segment OR. Point Y is obtained by rotating point M by angle pi/2 about point T. Point H is the reflection of point Y across segment TM. Let S be the midpoint of segment TM. Let C be the line through point O perpendicular to segment TM. Point V is the reflection of point R across line C. Let Q be the midpoint of segment MH. Point L is the reflection of point Q across point V. What is the length of segment SL? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let EO = 2. Point D is the midpoint of segment OE. Point S is obtained by rotating point E by angle 5pi/6 about point O. Point M is the reflection of point O across point E. Let K be the perpendicular bisector of segment OE. Let T be the reflection of point E across point M. Let V be the rotation of point E by angle 5pi/6 around point T. Point C is the reflection of point D across segment SE. Let N be the reflection of line K through point V. Point H is the reflection of point D across line N. Let P be the reflection of point C across point H. What is the length of segment PM? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
KV = 11. Let Z be the rotation of point K by angle 11pi/12 around point V. Let S be the perpendicular bisector of segment ZV. Let B be the reflection of point V across point Z. Let P be the reflection of line S through point B. Let J be the angle bisector of angle VZK. Point A is the reflection of point B across line J. Point W lies on line S. Let X be the reflection of point Z across point W. Let T be the midpoint of segment XA. Point E is the reflection of point T across line P. What is the distance between points W and E? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
XO = 5. Line B passes through points O and X. Let P be the reflection of point X across point O. Point W is the midpoint of segment OX. Point Z is the reflection of point W across point P. Line C passes through point X perpendicular to line B. Let E be the reflection of point Z across point X. Point Q lies on line C at distance 7 from point O. Point L is the reflection of point E across point W. Point Y is the reflection of point P across line C. Point T is the midpoint of segment YL. Point I is the reflection of point E across segment ZQ. Let M be the midpoint of segment IT. Find the distance between points M and X. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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DY = 1. Let B be the line through points Y and D. Let M be the reflection of point D across point Y. Point J is on line B. Point C is obtained by rotating point Y by angle pi/3 about point D. Let Q be the perpendicular bisector of segment MC. Point E is the reflection of point J across line Q. Line S is the perpendicular bisector of segment CE. Point W is the reflection of point M across line S. What is the distance between points W and E? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
FU = 11. Line X passes through points F and U. Point E is the reflection of point U across point F. Point J is the reflection of point U across point E. Point D is on line X. Let Y be the perpendicular bisector of segment UF. Point B is on line Y. Point M is the midpoint of segment BD. Let V be the midpoint of segment MF. Line G is the perpendicular bisector of segment BD. Line O passes through point D parallel to line G. Let C be the reflection of point J across point V. Point Q is the reflection of point C across line O. Let W be the midpoint of segment QM. What is the length of segment DW? Express your answer as a decimal number rounded to 2 decimal places.

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WR = 3. Point H is the reflection of point R across point W. Point P is the reflection of point H across point R. Let J be the angle bisector of angle PRH. Line F passes through point H perpendicular to segment WP. Point C lies on line J. Point U is the reflection of point C across line F. Find the length of segment CU. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let XI = 4. Point G is obtained by rotating point X by angle 11pi/12 about point I. Point B is the reflection of point X across point I. Point W is the midpoint of segment BG. T is constructed so that GT = 4. Point C is the midpoint of segment XG. Line P passes through points G and T. Point O is the midpoint of segment GC. Point Q is on line P. Point J is the midpoint of segment IO. Point E is the reflection of point W across segment QJ. Find the distance between points E and J. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let HQ = 6. Let B be the midpoint of segment QH. Point G is the midpoint of segment QB. Let U be the line through points B and G. Point T is obtained by rotating point G by angle 7pi/12 about point Q. Let W be the perpendicular bisector of segment BT. Point N is the reflection of point B across point T. Let A be the reflection of line W across line U. Lines U and A intersect at point X. Line K passes through points B and N. Let Y be the midpoint of segment GT. Point O is the reflection of point X across line K. What is the length of segment YO? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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LF = 6. Let J be the rotation of point F by angle 2pi/3 around point L. Let Q be the reflection of point F across point J. Line O passes through point Q parallel to segment LF. Let N be the line through point Q perpendicular to line O. Point T is the reflection of point F across line O. Point U is the reflection of point L across point Q. Let X be the reflection of line N through point U. Point H is the intersection of lines X and O. What is the length of segment HT? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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RZ = 7. Let P be the rotation of point R by angle 5pi/12 around point Z. Point Q lies on segment ZP. Let X be the reflection of point Z across point R. Point J is the midpoint of segment ZP. Point G lies on segment XR. Let A be the midpoint of segment XR. Let T be the angle bisector of angle QGP. Point U is the midpoint of segment AJ. Let C be the midpoint of segment UP. Line I passes through point C parallel to line T. Let N be the angle bisector of angle URP. Point L is the reflection of point A across line N. Point E is the reflection of point L across line I. What is the distance between points C and E? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let OH = 1. Let F be the reflection of point O across point H. Let T be the rotation of point F by angle pi/6 around point O. Let E be the vector from point T to point F. Let W be the midpoint of segment FT. Line S is the perpendicular bisector of segment FW. Point P is the reflection of point T across line S. Line U is the angle bisector of angle TOP. Point I is the reflection of point T across line U. Let X be point F translated by vector E. Point Y is the midpoint of segment PX. Point B is the reflection of point W across segment OH. Point C is the reflection of point Y across point B. What is the length of segment CI? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let HC = 2. Let Q be the vector from point C to point H. Let S be the perpendicular bisector of segment CH. Let G be the rotation of point H by angle 5pi/12 around point C. Let I be the rotation of point G by angle 5pi/6 around point H. Point J is the reflection of point I across line S. Let R be point J translated by vector Q. Let T be the rotation of point G by angle 5pi/6 around point I. Let X be the angle bisector of angle CTR. Point D is the reflection of point R across line X. Point Y is the midpoint of segment HD. Point B is the reflection of point T across point I. Let U be the midpoint of segment BR. Find the length of segment YU. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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UE = 7. Let B be the line through points E and U. Point Z is on line B. Let A be the reflection of point Z across point U. Point F is the reflection of point A across point E. Find the distance between points F and Z. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
UX = 5. Point S is obtained by rotating point X by angle 7pi/12 about point U. Point E is the reflection of point X across point U. Let V be the vector from point S to point U. Point Y is obtained by rotating point U by angle 11pi/12 about point E. Let Q be point U translated by vector V. Point J is the midpoint of segment YS. Point P is the reflection of point Q across point J. Point A is the reflection of point P across point U. Point G is the reflection of point X across segment YP. What is the length of segment GA? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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ES = 3. Line P passes through point S perpendicular to segment ES. Point L is the midpoint of segment SE. Point Y is the reflection of point L across line P. What is the distance between points Y and E? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let TB = 5. Let X be the reflection of point B across point T. Let D be the perpendicular bisector of segment BX. Point K is on line D with |KB| = 6. Point C is the reflection of point B across point X. Point J is the reflection of point T across segment KC. Let L be the midpoint of segment JX. Find the distance between points L and J. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let LY = 3. Let Q be the perpendicular bisector of segment YL. Point B lies on line Q at distance 11 from point L. Let H be the reflection of point B across point Y. Point U is the midpoint of segment BY. Line F passes through points L and U. Point M is the reflection of point H across line Q. Line S is the reflection of line Q across point M. Point W is the intersection of lines F and S. Segments MY and HL intersect at point E. Point K is the midpoint of segment MW. Find the length of segment KE. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let QY = 8. Let E be the perpendicular bisector of segment QY. Point J is the midpoint of segment QY. Point K is on line E with |KJ| = 12. Let L be the angle bisector of angle QYK. Let B be the midpoint of segment JK. Line S is the reflection of line L across point J. Let V be the midpoint of segment BQ. Point C is the reflection of point K across line S. Let X be the midpoint of segment VQ. Find the length of segment CX. Express your answer as a decimal number rounded to 2 decimal places.

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Let WH = 2. Point O is obtained by rotating point H by angle 7pi/12 about point W. Line Y is the angle bisector of angle WHO. Line I is the perpendicular bisector of segment WO. Let F be the reflection of point W across point H. Line G is the reflection of line Y across point W. Point D is obtained by rotating point H by angle 5pi/12 about point F. Point K is the reflection of point F across line I. Let X be the reflection of point D across point K. Point L is the reflection of point K across segment HF. Let V be the rotation of point X by angle 3pi/4 around point F. Point T is the reflection of point L across line Y. Let N be the midpoint of segment HF. Let P be the line through point N parallel to line G. Point E is the reflection of point T across line P. Find the length of segment EV. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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LI = 4. Point Z is the reflection of point I across point L. Point A is obtained by rotating point L by angle 2pi/3 about point I. Line Y passes through point I perpendicular to segment LZ. Point F is the reflection of point Z across line Y. Let H be the perpendicular bisector of segment FL. Line O passes through point F perpendicular to segment LZ. Let C be the line through point L parallel to line H. Let N be the reflection of point A across point Z. Let X be the reflection of line O across line C. Let Q be the line through point Z parallel to line Y. Segment NL and line X intersect at point J. Point E is the reflection of point A across line Q. What is the length of segment EJ? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


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GV = 3. Point B is the reflection of point G across point V. Let C be the reflection of point B across point G. Let J be the reflection of point G across point C. Let Z be the line through points J and C. Let H be the line through point B perpendicular to line Z. Point F is the midpoint of segment GV. Let R be the midpoint of segment FG. Let M be the reflection of point C across point F. Point L is the reflection of point C across line H. Point I is the reflection of point R across point M. Point T is the reflection of point L across point G. What is the distance between points I and T? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let AO = 5. Let B be the midpoint of segment AO. Point S is the midpoint of segment BA. Point U is obtained by rotating point A by angle 5pi/12 about point S. Let F be the reflection of point U across point O. Point W is the midpoint of segment UO. Let Z be the reflection of point A across point W. Let H be the reflection of point F across point S. Find the distance between points Z and H. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let XO = 10. Let G be the reflection of point O across point X. Point B is the midpoint of segment XG. N is constructed so that BN = 3. Let L be the line through points O and B. Let Q be the reflection of point B across point N. Point W is on line L. Point F is the reflection of point Q across segment WN. Point J is the midpoint of segment NF. Point A is the midpoint of segment NJ. What is the length of segment JA? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
EH = 12. Let T be the rotation of point H by angle pi/3 around point E. Point I is obtained by rotating point E by angle 3pi/4 about point T. Let Y be the line through points I and T. Line S is the reflection of line Y across point E. Point Z is the reflection of point E across line S. Point F is the reflection of point H across line S. Point J is the reflection of point F across line Y. Point X is the reflection of point E across point Z. Line Y and segment JZ intersect at point C. Point K is the reflection of point C across point Z. Let N be the midpoint of segment XK. What is the distance between points I and N? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
PN = 11. Point A is obtained by rotating point P by angle pi about point N. Point X is obtained by rotating point P by angle pi/12 about point N. Let I be the rotation of point P by angle pi/4 around point A. Point Q is the midpoint of segment AN. Line Z passes through point P perpendicular to segment NP. Line C passes through point Q parallel to line Z. Point J is the reflection of point I across line C. Point E is the reflection of point X across line C. Point F is the midpoint of segment NP. Let Y be the perpendicular bisector of segment JE. Line V is the reflection of line Z across line Y. Lines V and C intersect at point H. Point T is the reflection of point H across segment NP. What is the distance between points F and T? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let OT = 9. Let E be the midpoint of segment TO. Point G is obtained by rotating point E by angle pi/3 about point O. Let V be the angle bisector of angle OEG. Let Q be the reflection of point T across point G. Point X is the reflection of point Q across line V. Let I be the midpoint of segment EX. Point D is the midpoint of segment IE. What is the distance between points D and X? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
FA = 5. Let Y be the perpendicular bisector of segment FA. Let L be the reflection of point A across point F. Line B passes through point L parallel to line Y. Let G be the reflection of line Y across line B. Let S be the line through point A parallel to line G. Let J be the perpendicular bisector of segment FL. Let H be the reflection of line S across line J. Point O is the reflection of point F across line H. Segment OF and line G intersect at point M. What is the distance between points O and M? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}