[User]
SM = 12. Let V be the midpoint of segment SM. Point A is obtained by rotating point V by angle 11pi/12 about point M. SX = 4. Let K be the reflection of point V across point A. Let Z be the reflection of point V across point X. Let R be the angle bisector of angle XZA. Point W lies on line R at distance 2 from point S. Point J is the midpoint of segment MK. Let E be the line through point W parallel to segment MK. Line H passes through point A perpendicular to line E. Line I passes through point J parallel to line H. Point Y is the reflection of point S across line I. What is the length of segment YK? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PS = 2. Let Z be the reflection of point S across point P. Point N is obtained by rotating point P by angle 7pi/12 about point Z. Point I is the midpoint of segment SP. Let D be the rotation of point I by angle pi/2 around point N. Line E is the angle bisector of angle SNP. Point V is the midpoint of segment SD. Line U is the perpendicular bisector of segment DI. Point F is the reflection of point P across point N. Line H passes through point I parallel to line E. Let G be the rotation of point S by angle 2pi/3 around point I. Let T be the line through points G and V. Let Q be the reflection of line U across line T. Let K be the line through point Z parallel to line Q. Point R is the intersection of lines H and K. Point O is the reflection of point F across point D. What is the distance between points O and R? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let CL = 10. Point Q is the midpoint of segment LC. Let M be the rotation of point Q by angle pi/3 around point L. Point E is obtained by rotating point L by angle pi/4 about point Q. Let H be the vector from point E to point L. Let V be the perpendicular bisector of segment QE. Point W is the reflection of point M across line V. Point T is the translation of point W by vector H. Point Z is the midpoint of segment CT. Point F is the midpoint of segment TZ. Let Y be the midpoint of segment FQ. Point D is the reflection of point Y across segment LW. Find the distance between points D and C. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KM = 1. Let E be the rotation of point K by angle 2pi/3 around point M. Let S be the reflection of point E across point K. Let X be the reflection of point M across point K. Line P is the angle bisector of angle MSK. Point A is the reflection of point X across line P. Point B is the reflection of point A across segment EX. Point G is the reflection of point M across segment EX. Point F is the reflection of point G across point B. Find the distance between points F and E. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YO = 4. Let X be the rotation of point Y by angle 5pi/6 around point O. Point K is obtained by rotating point X by angle pi/6 about point Y. Let Q be the perpendicular bisector of segment OY. Point D is the reflection of point X across line Q. Let E be the midpoint of segment DO. Point F is the midpoint of segment EK. What is the length of segment DF? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let TY = 8. Let I be the vector from point T to point Y. Let D be point Y translated by vector I. Line M is the perpendicular bisector of segment TD. Point O is the midpoint of segment YT. Point J is the reflection of point D across point O. Point X is the reflection of point J across point Y. Let C be the perpendicular bisector of segment JY. Point W is on line C with |WX| = 8. Line B is the reflection of line C across line M. Let E be the line through point J parallel to line B. Point V is the reflection of point Y across line E. Point S is the midpoint of segment VW. What is the distance between points V 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 WE = 5. Line S is the perpendicular bisector of segment WE. Point I is the midpoint of segment WE. Point R is obtained by rotating point W by angle 11pi/12 about point E. Line K passes through points R and W. Point L is the reflection of point I across line K. Line D passes through point L parallel to line S. Let Y be the reflection of line D through point I. Point C is the reflection of point R across line Y. Find 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 NA = 7. Point K is the midpoint of segment NA. Line U is the perpendicular bisector of segment AK. Let B be the line through point N parallel to line U. Point R is the reflection of point K across line B. Find the distance between points K and R. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RV = 11. Point D is obtained by rotating point R by angle pi/12 about point V. Let T be the midpoint of segment RV. Line C is the angle bisector of angle DTV. Let Y be the rotation of point D by angle pi/2 around point V. Line B passes through point V perpendicular to line C. Let K be the reflection of point Y across point R. Let L be the reflection of line B through point T. Point Q is the reflection of point D across line L. Let I be the reflection of point K across point Q. What is the distance between points K 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 QF = 9. Let X be the midpoint of segment FQ. Line I passes through points X and Q. Point L is the midpoint of segment XQ. Let G be the reflection of point Q across point F. Point D is obtained by rotating point F by angle pi/2 about point L. Let K be the perpendicular bisector of segment XF. Point A is the reflection of point X across point D. Line P passes through points A and Q. Let U be the reflection of line P across line K. Point H is the midpoint of segment DG. Line C is the reflection of line I across point H. Point S is the intersection of lines C and U. What is the length of segment SH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KO = 9. Point I is obtained by rotating point K by angle 3pi/4 about point O. Let R be the midpoint of segment OK. Line G is the perpendicular bisector of segment KO. Point A is the midpoint of segment OI. Let Q be the line through point A parallel to line G. Point B is obtained by rotating point I by angle pi/2 about point A. Let X be the reflection of line G across line Q. Point P is the reflection of point I across point B. Point F is the reflection of point R across line X. Let Y be the perpendicular bisector of segment IK. Let T be the line through point P parallel to line Y. Point N is the reflection of point R across line T. Let D be the midpoint of segment NF. What is the distance between points D and P? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JM = 11. Point F is the midpoint of segment MJ. Let I be the reflection of point F across point M. Point D is the midpoint of segment IM. Point C is the reflection of point D across point I. Find the distance between points D and C. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
UB = 11. Let E be the rotation of point B by angle 5pi/6 around point U. Let F be the midpoint of segment BU. Point N is the midpoint of segment EF. Let H be the rotation of point B by angle pi around point E. Point I is the reflection of point N across segment FB. Let V be the vector from point F to point U. Let Y be point I translated by vector V. Line C is the perpendicular bisector of segment FB. Let X be the line through point Y parallel to line C. Point G is the reflection of point N across line X. Point D is the reflection of point H across point Y. Point A is the midpoint of segment IG. Find the length of segment DA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LR = 1. Point Y is the reflection of point L across point R. Point I is the reflection of point L across point Y. Point M is the midpoint of segment RL. Point E is the reflection of point M across point I. Find the distance between points L 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 CN = 11. Let L be the reflection of point C across point N. Let W be the rotation of point N by angle pi/12 around point C. Let V be the perpendicular bisector of segment WL. Let Y be the midpoint of segment CW. Let S be the reflection of point N across point W. Line U is the reflection of line V across point N. Point K is the intersection of line U and segment SY. What is the length of segment KL? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GI = 8. Let F be the midpoint of segment GI. Let X be the rotation of point F by angle pi/12 around point I. Point K is obtained by rotating point F by angle 11pi/12 about point X. Let Y be the angle bisector of angle IXK. Point W is the midpoint of segment GK. Let E be the line through point K parallel to line Y. Point R is the intersection of segment GI and line E. Find the distance between points R and W. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XH = 10. Let M be the midpoint of segment HX. Let B be the rotation of point M by angle pi around point H. Let I be the rotation of point H by angle pi/2 around point B. Line J passes through points I and X. Point D is the midpoint of segment MX. Point S is the reflection of point B across line J. Let Z be the reflection of point D across point I. Find the distance between points S and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QH = 11. Let D be the line through points H and Q. Point C is the reflection of point H across point Q. Let R be the vector from point Q to point C. Let J be point C translated by vector R. Let Z be the rotation of point H by angle pi/6 around point J. Let W be the angle bisector of angle HCZ. Point E is the reflection of point Z across point J. Line X passes through points E and C. Line U is the reflection of line X across line D. Point P is the reflection of point Z across line W. Point N is the reflection of point P across line U. What is the length of segment ZN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RQ = 7. Point T is obtained by rotating point Q by angle pi/3 about point R. Point Y is the reflection of point R across point Q. Point B is the midpoint of segment YQ. Let S be the rotation of point R by angle pi/3 around point Y. Let E be the reflection of point S across point Q. Let D be the midpoint of segment TB. Let J be the midpoint of segment DR. Let C be the angle bisector of angle TJB. Let W be the line through point E parallel to line C. Point A is the reflection of point B across segment QT. Point F is the reflection of point R across line W. Find the length of segment FA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PU = 11. Let B be the line through points U and P. Let S be the reflection of point P across point U. Point F is obtained by rotating point U by angle pi/2 about point P. Let L be the perpendicular bisector of segment SF. Line D is the reflection of line L across line B. Let W be the rotation of point U by angle 11pi/12 around point S. Let K be the reflection of point P across point S. Point T is the reflection of point W across segment KF. Point E is the intersection of segment KF and line D. Find the distance between points T and E. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XT = 1. Let H be the perpendicular bisector of segment TX. Let D be the midpoint of segment TX. Point P lies on line H at distance 7 from point X. Line I is the angle bisector of angle TDP. Point Z is the reflection of point X across line I. Let U be the midpoint of segment ZP. Let C be the midpoint of segment TD. Let J be the midpoint of segment UD. Let R be the reflection of point C across point U. Let A be the line through point R parallel to line I. Point B is the intersection of line A and segment PT. What is the length of segment BJ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EJ = 6. Point H is obtained by rotating point J by angle 5pi/12 about point E. Let N be the rotation of point J by angle pi around point E. Point I is the reflection of point H across segment NJ. Point Z is the midpoint of segment JI. Point V is the reflection of point E across point I. Let D be the reflection of point Z across point V. Point K is the reflection of point D across point I. What is the distance between points V and K? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IE = 4. Let P be the reflection of point I across point E. Let A be the rotation of point I by angle 11pi/12 around point E. Let F be the vector from point I to point A. Let V be the perpendicular bisector of segment IP. Point B is the translation of point E by vector F. Point U is obtained by rotating point B by angle pi about point I. Point X is the reflection of point A across line V. Point M is the midpoint of segment XI. Point Y is the midpoint of segment UM. Let W be the reflection of point Y across point E. Find the distance between points W and M. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ZH = 2. Point Q is the midpoint of segment HZ. Let N be the midpoint of segment HQ. Let O be the midpoint of segment QN. Point P is obtained by rotating point Z by angle 3pi/4 about point N. Let T be the reflection of point O across point N. Point D is the reflection of point Q across point H. Let V be the line through points O and N. Line F passes through point D perpendicular to line V. Let J be the reflection of line F through point P. Let A be the line through points P and N. Let K be the reflection of point T across point O. Point E is the intersection of lines A and J. Point S is the midpoint of segment KE. What is the distance between points T and S? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IM = 4. Line F passes through points I and M. Let IL = 11. Point D is the reflection of point L across line F. Line H is the angle bisector of angle DIM. Point Z is the reflection of point D across line H. Let T be the reflection of point Z across point I. Line P passes through point T perpendicular to line F. Point K is the reflection of point Z across line P. Point B is the midpoint of segment KI. 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]
Let ZP = 11. Line O is the perpendicular bisector of segment PZ. Point R is the reflection of point P across point Z. Point H is the reflection of point R across line O. Point E is on line O with |EP| = 4. Let Y be the vector from point Z to point E. Point I is the translation of point P by vector Y. Let M be the line through point H perpendicular to segment EI. Point V is the reflection of point I across segment EP. Point U is the reflection of point Z across line M. Let D be the midpoint of segment VU. Find the length of segment ZD. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
MZ = 11. Let X be the midpoint of segment ZM. Let G be the reflection of point M across point Z. Let O be the midpoint of segment XZ. Point L is the midpoint of segment ZG. What is the distance between points O and L? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FH = 8. Let L be the rotation of point H by angle 5pi/12 around point F. Line D passes through points H and F. Let I be the midpoint of segment HL. Point J is the reflection of point L across line D. Point N is the reflection of point J across segment IL. Point X is the midpoint of segment IL. Find 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}


[User]
MU = 3. Let E be the rotation of point U by angle 7pi/12 around point M. Point Q is the midpoint of segment UM. Point R is obtained by rotating point E by angle 7pi/12 about point U. Line W passes through point R parallel to segment UQ. Line Z is the reflection of line W across point U. Let N be the midpoint of segment UE. Point H is the reflection of point U across line Z. What is the distance between points H 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 DZ = 4. Point T is the midpoint of segment DZ. Point E is the reflection of point Z across point D. Line J is the angle bisector of angle DTE. Line H passes through point Z perpendicular to line J. Let Q be the rotation of point T by angle pi/12 around point D. Let L be the angle bisector of angle QDE. Line A is the reflection of line H across line L. Let O be the line through points Q and E. Point U is the reflection of point E across line L. Point K is the intersection of lines O and A. Point V is the reflection of point K across point U. Let Y be the midpoint of segment QT. Find the distance between points V and Y. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LG = 12. Line V is the perpendicular bisector of segment GL. Point S is on line V. Line X passes through point S perpendicular to line V. Let P be the rotation of point G by angle 7pi/12 around point L. Point A is the reflection of point P across segment GL. Line W passes through point G perpendicular to line X. Point K is the reflection of point A across line W. Let B be the reflection of point P across point K. Let M be the reflection of point K across point B. Point C is the reflection of point M across segment PS. What is the distance between points C and P? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EO = 4. Let R be the perpendicular bisector of segment OE. Point Y lies on line R at distance 11 from point O. Let K be the midpoint of segment EY. Let C be the reflection of point Y across point E. Point T is the reflection of point K across point O. Let L be the midpoint of segment YO. Let D be the line through points C and T. Point B is the reflection of point L across line D. Point Z is the reflection of point L across point B. Find the distance between points Z 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 DY = 11. Let H be the vector from point D to point Y. Let U be the perpendicular bisector of segment YD. Point M lies on segment YD. Line X is the perpendicular bisector of segment YM. Let E be the line through point Y perpendicular to segment YD. Line L passes through point D parallel to line E. Line S is the reflection of line U across point Y. Let F be the reflection of point D across point Y. Point A is the midpoint of segment YD. Line B is the reflection of line S across line L. Point P is the midpoint of segment AY. Let G be the line through point P parallel to line X. Let Z be the rotation of point P by angle 7pi/12 around point D. Point O is the reflection of point P across line S. Point J is on line B with |JO| = 8. Let W be the reflection of line G through point J. Let N be point P translated by vector H. Point V is the reflection of point Y across point D. Point K lies on line W at distance 3 from point V. Point T is the reflection of point V across segment NK. Let Q be the line through point Z parallel to line U. Point I is the intersection of segment YT and line Q. What is the length of segment IF? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let IZ = 4. Let P be the rotation of point Z by angle 2pi/3 around point I. Let E be the midpoint of segment ZP. Point K is obtained by rotating point P by angle pi about point I. Let T be the rotation of point I by angle 5pi/6 around point E. Point F is the midpoint of segment TK. Let O be the midpoint of segment IF. Let A be the angle bisector of angle IZO. Let G be the reflection of point P across point F. Point Q is the reflection of point G across line A. What is the distance between points O 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 MK = 6. Line S is the perpendicular bisector of segment KM. Let F be the line through point K parallel to line S. Point D is the midpoint of segment KM. Point N is the reflection of point M across line F. Point Q is the reflection of point D across point M. Let E be the rotation of point N by angle pi/2 around point Q. Point U is obtained by rotating point K by angle pi/12 about point N. Let A be the line through point U perpendicular to line F. Line J is the reflection of line A across point Q. Point L is the reflection of point K across line J. Let Y be the reflection of point E across point N. What is the distance between points L 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 TK = 9. Point Z is the midpoint of segment TK. Point I is obtained by rotating point T by angle pi/6 about point K. Point L is the midpoint of segment IZ. Let R be the rotation of point T by angle 3pi/4 around point Z. Line O passes through points I and T. Let N be the reflection of point T across point I. Let Y be the midpoint of segment NR. Line M is the reflection of line O across point Y. Point S is the reflection of point I across line M. Find 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 DL = 11. Line J is the perpendicular bisector of segment DL. Let H be the midpoint of segment LD. Point B is on line J with |BL| = 4. Let O be the vector from point D to point L. Point W is the reflection of point D across point L. Let E be the line through point W parallel to line J. Point I is the translation of point H by vector O. Point P is the reflection of point D across line E. Line N passes through points I and B. Point K is the midpoint of segment PW. Let C be the reflection of line E across line N. Line M passes through point B perpendicular to line J. Point Y is the intersection of segment KB and line C. Point S is the reflection of point W across line M. What is the distance between points S and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VO = 11. Line M is the perpendicular bisector of segment OV. Let F be the reflection of line M through point V. Line Y passes through point O parallel to line F. Point H is obtained by rotating point O by angle 5pi/6 about point V. Let R be the reflection of point H across point O. Let P be the midpoint of segment OR. Line C is the perpendicular bisector of segment RO. Let S be the reflection of point V across point P. Let E be the reflection of line M across line C. Point G is the reflection of point P across point S. Lines E and Y intersect at point Z. What is the distance between points Z 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 RL = 7. Let J be the reflection of point R across point L. Line U is the perpendicular bisector of segment LR. Let M be the reflection of line U through point J. Line E passes through points R and L. Point N is the intersection of lines E and M. Point I is the midpoint of segment JN. Let H be the midpoint of segment JI. Find the distance between points H 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 TO = 7. Let A be the reflection of point O across point T. Let E be the vector from point T to point O. Let D be the midpoint of segment TO. Point G is obtained by rotating point O by angle 11pi/12 about point D. Let C be point D translated by vector E. Point U is the reflection of point A across point D. Let F be the rotation of point U by angle 7pi/12 around point C. Let Q be the angle bisector of angle ADG. Line I is the angle bisector of angle OAG. Let V be the reflection of point A across point O. Point J is the reflection of point C across line I. Point L is the reflection of point J across line Q. Point M is the midpoint of segment FV. What is the distance between points M and L? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
UR = 5. Let A be the vector from point U to point R. Point Z is the midpoint of segment UR. Let T be point Z translated by vector A. Let O be the rotation of point R by angle pi around point T. Let J be the perpendicular bisector of segment OZ. Let B be the rotation of point U by angle pi/4 around point T. Line C passes through points O and B. Let Q be the reflection of line C across line J. Lines Q and C intersect at point D. Let V be the midpoint of segment TO. Find the length of segment VD. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OP = 7. Point Z is the reflection of point P across point O. Point Q is obtained by rotating point Z by angle pi/3 about point O. Let M be the reflection of point P across point Q. Line C is the perpendicular bisector of segment PQ. Point T is on line C. Let V be the rotation of point T by angle pi/2 around point Q. Point Y is the reflection of point M across point V. Line K passes through point Y perpendicular to segment MP. Line E passes through point V parallel to line K. Point X is the reflection of point Y across line E. What is the length of segment XZ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KG = 4. Point N is the reflection of point G across point K. Point Z is the midpoint of segment KN. Point J is obtained by rotating point N by angle pi/2 about point Z. Let E be the rotation of point K by angle pi/4 around point G. Line L passes through point Z parallel to segment JE. Line L and segment GJ intersect at point A. What is the length of segment NA? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
CZ = 4. M is placed 10 units away from Z. Let V be the reflection of point M across point C. Let Q be the midpoint of segment ZM. Let E be the midpoint of segment ZV. Let R be the midpoint of segment VQ. Find the length of segment ER. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let YQ = 7. Point T is obtained by rotating point Y by angle 7pi/12 about point Q. Line O is the angle bisector of angle TYQ. Point R is the midpoint of segment QT. Point Z is the reflection of point R across point T. Point E is the reflection of point R across line O. Let F be the reflection of point E across point Z. Find the length of segment FT. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let XQ = 3. Let E be the midpoint of segment QX. Point P is the midpoint of segment XE. Let V be the vector from point Q to point E. Point U is obtained by rotating point X by angle 2pi/3 about point Q. Point J is obtained by rotating point X by angle pi/6 about point U. Let B be the reflection of point P across point U. Point O is the translation of point U by vector V. Let F be the reflection of point B across point O. Point L is the midpoint of segment PQ. Point S is the reflection of point L across point F. Line N is the angle bisector of angle BFJ. Point Z is the reflection of point O across line N. Find the distance between points S and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FR = 12. Let G be the line through points F and R. Point D is on line G with |DF| = 12. Let B be the midpoint of segment DF. Let C be the rotation of point B by angle pi/3 around point D. Let K be the rotation of point B by angle pi/12 around point D. Point L is obtained by rotating point C by angle 7pi/12 about point B. Let H be the midpoint of segment KC. Let X be the reflection of point H across point C. Point N is the midpoint of segment HB. Point T is the reflection of point X across point N. Point M is the reflection of point L across point T. What is the length of segment XM? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LR = 8. Let H be the midpoint of segment RL. Let W be the angle bisector of angle HLR. Point F is obtained by rotating point H by angle 5pi/6 about point L. Let M be the reflection of line W through point F. Let K be the rotation of point H by angle pi/3 around point R. Let V be the reflection of line W across line M. Point D is the midpoint of segment KL. Line T passes through points R and D. Lines V and T intersect at point A. Let J be the reflection of point K across point A. Find the distance between points J and D. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OH = 12. Point A is obtained by rotating point H by angle 11pi/12 about point O. Point W is obtained by rotating point O by angle 5pi/12 about point A. Point N is the midpoint of segment AH. Point X is the reflection of point W across segment AH. Let E be the reflection of point N across point A. Find the distance between points X 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 IV = 8. Let R be the reflection of point I across point V. Let J be the perpendicular bisector of segment VR. Point N is the reflection of point I across line J. Let G be the midpoint of segment IV. Find the distance between points G and N. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EP = 4. Let H be the reflection of point P across point E. Point C is obtained by rotating point H by angle 5pi/12 about point E. Line T is the angle bisector of angle CEP. Line J is the reflection of line T across point C. Point U is on line J with |UC| = 12. Let W be the midpoint of segment CE. Point D is the reflection of point U across point W. Point I is the reflection of point H across segment CE. Point Z is the midpoint of segment EI. Let Q be the midpoint of segment DI. What is the length of segment ZQ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let EZ = 9. Let A be the rotation of point E by angle pi/12 around point Z. Let X be the vector from point Z to point E. Let T be the angle bisector of angle ZEA. Let W be the reflection of line T through point A. Let K be the perpendicular bisector of segment AZ. Line N passes through point E perpendicular to line T. Let C be the reflection of line K across line W. Let J be point E translated by vector X. Let Y be the line through point J parallel to line N. Point B is the reflection of point E across line Y. Line S is the reflection of line N across point A. Point U is the reflection of point B across line S. Point V is the intersection of lines C and Y. Find the length of segment VU. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SG = 12. Let W be the midpoint of segment GS. Let X be the rotation of point W by angle pi/6 around point S. Point T is obtained by rotating point S by angle 2pi/3 about point G. Let V be the midpoint of segment SX. Let A be the reflection of point T across point G. Point J is the midpoint of segment VA. Point U is the midpoint of segment SJ. What is the length of segment JU? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EM = 1. Point D is the reflection of point E across point M. Point I is obtained by rotating point M by angle pi/6 about point E. Point N is the reflection of point D across point I. Let B be the line through points N and E. Let A be the midpoint of segment MI. Let F be the line through point A parallel to segment MD. Lines F and B intersect at point Q. Let G be the line through point Q perpendicular to segment MD. Let W be the reflection of line B across line G. Point V is the reflection of point D across line W. Find the length of segment IV. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GQ = 8. Let V be the rotation of point Q by angle 7pi/12 around point G. Let N be the reflection of point Q across point G. Let P be the rotation of point Q by angle 5pi/12 around point G. Let S be the angle bisector of angle PVN. Point E is the reflection of point G across line S. Point I is the midpoint of segment VN. Point C is the midpoint of segment IP. Point K is the midpoint of segment EP. Point W is the intersection of segments CN and KI. Find the length of segment PW. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EQ = 10. Point O is obtained by rotating point Q by angle 2pi/3 about point E. Point K is the reflection of point Q across point E. Point V is the reflection of point O across point E. Point D is the midpoint of segment KO. Let A be the line through point V perpendicular to segment VD. Point L is the reflection of point D across line A. What is the distance between points D and L? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let QW = 3. Let X be the reflection of point Q across point W. Let M be the midpoint of segment XW. Point J is obtained by rotating point Q by angle pi/6 about point M. Line F is the angle bisector of angle XWQ. Let B be the midpoint of segment WM. Point I is the reflection of point J across line F. Let C be the reflection of point B across point I. 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}


[User]
Let TC = 2. Point Y is the reflection of point C across point T. Let E be the line through points T and C. Line H passes through point C perpendicular to line E. Point M is obtained by rotating point C by angle pi/12 about point T. Let W be the midpoint of segment MY. Point B is the reflection of point M across point C. Point V is the reflection of point B across line H. Point G is the midpoint of segment VW. Find the distance between points B and G. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
DF = 6. Point T is the midpoint of segment DF. Let J be the midpoint of segment TD. Point P is the midpoint of segment FJ. Let N be the rotation of point F by angle pi/4 around point T. Point W is obtained by rotating point T by angle 5pi/12 about point P. Let Q be the midpoint of segment PN. Point L is the reflection of point Q across point J. Let A be the reflection of point J across point L. Point R is the reflection of point L across point A. Point B is the reflection of point L across segment TW. What is the distance between points B and R? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BW = 11. Let P be the midpoint of segment WB. Let Z be the rotation of point P by angle 11pi/12 around point B. Let H be the angle bisector of angle ZWB. Point E is obtained by rotating point P by angle 3pi/4 about point Z. Point T is on line H. Point I is the midpoint of segment BT. Point O is the reflection of point P across line H. Point G is the reflection of point O across point I. Let A be the reflection of line H through point G. Point Y is the reflection of point B across segment EZ. Point U is the reflection of point W across line A. 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}


[User]
Let SM = 5. Let A be the rotation of point M by angle pi/3 around point S. Let W be the vector from point A to point M. Point B is the midpoint of segment SM. Point U is obtained by rotating point B by angle 5pi/12 about point S. Let F be point M translated by vector W. Let Q be the perpendicular bisector of segment MS. Point P is obtained by rotating point M by angle 7pi/12 about point B. Point L is the reflection of point P across point F. Let J be the midpoint of segment BM. Point E is the reflection of point L across line Q. Let Y be the midpoint of segment JU. What is the distance between points Y 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 PM = 12. Line A is the perpendicular bisector of segment MP. Let B be the line through point M parallel to line A. Let L be the reflection of line B across line A. Point S is the reflection of point M across line L. Let W be the midpoint of segment SP. Let E be the rotation of point W by angle 5pi/6 around point M. Point X is the midpoint of segment EP. Let Y be the midpoint of segment XW. Point U is the midpoint of segment MY. Find the distance between points U and Y. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VS = 10. Line M is the perpendicular bisector of segment VS. Point J is obtained by rotating point V by angle pi/4 about point S. Let X be the rotation of point S by angle pi/6 around point V. Let I be the rotation of point S by angle pi/3 around point X. Let Y be the reflection of point I across point V. Let D be the reflection of point J across point X. Line N is the reflection of line M across point D. Point K is the midpoint of segment YS. Point W is the reflection of point Y across line N. Let C be the midpoint of segment IK. What is the length of segment WC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BN = 12. Point K is obtained by rotating point B by angle pi/3 about point N. Line J is the angle bisector of angle NBK. Point T is the reflection of point B across point N. Let C be the reflection of line J through point T. Point X is the reflection of point N across line C. Let Q be the midpoint of segment KX. Find the length of segment TQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VS = 11. Let X be the midpoint of segment VS. Point H is obtained by rotating point X by angle pi/6 about point S. Point O is the midpoint of segment VH. Point M is obtained by rotating point O by angle 3pi/4 about point X. Point Z is obtained by rotating point X by angle pi/3 about point S. Let Q be the line through points Z and M. Let N be the line through point X perpendicular to line Q. Point I is the reflection of point Z across line N. Let U be the midpoint of segment ZS. What is the distance between points U and I? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VO = 5. Point U is the midpoint of segment VO. Let C be the rotation of point V by angle pi/4 around point U. Let P be the rotation of point C by angle 3pi/4 around point O. Point F is obtained by rotating point C by angle pi/12 about point V. Line T is the angle bisector of angle PCF. Let S be the perpendicular bisector of segment VP. Point A is the reflection of point O across line S. Line Z is the reflection of line S across line T. Let D be the midpoint of segment AP. Point M is the intersection of segment FA and line Z. Point Y is the reflection of point P across segment FA. Point L is the reflection of point D across point M. Let W be the midpoint of segment YV. What is the length of segment WL? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PJ = 9. Point M is obtained by rotating point J by angle pi/2 about point P. Let T be the perpendicular bisector of segment PJ. Let L be the line through point M parallel to line T. Let W be the midpoint of segment MJ. Line R passes through point W parallel to segment PJ. Point O is the intersection of lines R and L. What is the measure of angle PWO, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JH = 8. Let F be the rotation of point H by angle 5pi/6 around point J. Point K is obtained by rotating point J by angle 5pi/6 about point F. Point M is obtained by rotating point K by angle pi/3 about point H. Point D is the midpoint of segment HM. Let T be the reflection of point F across point D. Point Y is the reflection of point T across point K. What is the distance between points D and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AM = 9. Point O is obtained by rotating point M by angle 7pi/12 about point A. AL = 9. Let F be the rotation of point A by angle 11pi/12 around point O. Let T be the line through points A and O. Line P passes through point F parallel to line T. Point K is the reflection of point A across line P. Point H is the reflection of point L across line P. Find the distance between points H and K. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
MP = 1. Line J is the perpendicular bisector of segment PM. Point D is on line J with |DP| = 2. Point O is obtained by rotating point P by angle pi/3 about point D. Let G be the vector from point P to point O. Let N be the reflection of point O across point P. Let Q be the line through points N and M. Point H is the translation of point D by vector G. Let A be the rotation of point H by angle pi around point O. Let W be the midpoint of segment AM. Point L is the reflection of point H across line Q. Let K be the perpendicular bisector of segment LW. Line F is the reflection of line K across point P. Point S is the reflection of point A across line F. What is the length of segment SW? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QS = 2. Let V be the rotation of point Q by angle 7pi/12 around point S. Point D is the reflection of point V across segment QS. Point Z is the reflection of point D across point S. Point C is the midpoint of segment QS. Let X be the midpoint of segment CZ. Point J is the reflection of point X across segment QS. Find the distance between points J and X. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YE = 6. Let T be the reflection of point E across point Y. Let W be the reflection of point E across point T. Point R is obtained by rotating point Y by angle 3pi/4 about point W. Line C passes through points R and W. Let K be the perpendicular bisector of segment TE. Line L is the reflection of line K across line C. Line M passes through point T parallel to line C. Lines M and L intersect at point I. Find the length of segment IY. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let FQ = 12. Point I is obtained by rotating point Q by angle 2pi/3 about point F. Let U be the reflection of point F across point Q. Line E is the perpendicular bisector of segment FI. Point G is the reflection of point Q across line E. Line Z is the angle bisector of angle UFQ. Let V be the rotation of point F by angle pi/4 around point U. Point H is the midpoint of segment VQ. Let L be the line through point U perpendicular to line Z. Let O be the rotation of point I by angle 11pi/12 around point V. Point M is the reflection of point O across line Z. Let T be the midpoint of segment HM. Line D passes through point T perpendicular to line L. Point X is the reflection of point Q across line D. Find the distance between points X 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 HF = 9. Line S is the perpendicular bisector of segment FH. Point I is obtained by rotating point H by angle pi about point F. Let R be the vector from point H to point I. Line G is the angle bisector of angle HFI. Let A be the midpoint of segment FH. Let J be point I translated by vector R. Point Y lies on line G at distance 2 from point H. Let N be the reflection of point Y across point F. Let D be the angle bisector of angle NJF. Line B is the reflection of line D across point A. Point P is the translation of point Y by vector R. Let Z be the reflection of line S through point F. Let K be the midpoint of segment IY. Point O is the intersection of lines B and Z. Point V is the reflection of point O across point P. What is the length of segment KV? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let AP = 5. Point M is the midpoint of segment AP. Let X be the midpoint of segment AM. Let O be the rotation of point X by angle pi/3 around point M. Line S is the perpendicular bisector of segment OX. Line W is the angle bisector of angle XAM. Point C is the reflection of point A across line S. Let T be the vector from point M to point P. Point Q is the translation of point O by vector T. Let H be the line through point P perpendicular to line W. Point I is the midpoint of segment MX. Let B be the reflection of point A across point I. Point Y is the reflection of point Q across line H. Point F is the reflection of point Y across point B. What is the distance between points F 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 RH = 2. Point S is the reflection of point H across point R. Let C be the midpoint of segment RS. Let I be the rotation of point R by angle 2pi/3 around point C. Let E be the reflection of point R across point S. Point G is the reflection of point I across point C. Let A be the midpoint of segment GC. Let N be the angle bisector of angle ECI. Let D be the midpoint of segment HI. Let Q be the line through point R parallel to line N. Segment AD and line Q intersect at point U. Find the length of segment GU. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let NY = 12. Point M is the reflection of point Y across point N. Point C is the midpoint of segment NM. Point J is the reflection of point M across point Y. Line O passes through point N perpendicular to segment NM. Point Q is obtained by rotating point Y by angle pi/3 about point C. Let Z be the line through point Q parallel to line O. Line H passes through point C perpendicular to segment NM. Point W is the reflection of point J across line H. Point D is the reflection of point W across line Z. Find the distance between points D 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 KW = 12. Line A is the perpendicular bisector of segment KW. Let P be the rotation of point K by angle pi/4 around point W. Line Z is the angle bisector of angle PKW. Line T is the reflection of line A across line Z. Line Y passes through point W parallel to line A. Let M be the midpoint of segment KP. Point J is the intersection of lines T and Y. What is the length of segment MJ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SE = 4. Point H is the reflection of point E across point S. Let I be the rotation of point S by angle 11pi/12 around point H. Point A is the midpoint of segment SE. Point C is the reflection of point I across segment HE. Point P is the midpoint of segment CA. Let T be the reflection of point P across point I. Point D is the midpoint of segment TI. What is 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]
IA = 4. Let G be the rotation of point I by angle 5pi/12 around point A. Point E is the midpoint of segment AI. Let L be the angle bisector of angle GAE. Point Q is obtained by rotating point G by angle pi/3 about point E. Let C be the rotation of point Q by angle pi/3 around point A. Line N is the perpendicular bisector of segment AI. Let S be the line through points E and C. Let Y be the line through point Q perpendicular to line L. Let R be the reflection of line N across line Y. Point K is the intersection of lines R and S. Point X is the reflection of point G across line R. What is the distance between points K and X? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KY = 5. Let O be the perpendicular bisector of segment KY. Line X passes through point K parallel to line O. Let D be the rotation of point Y by angle pi/12 around point K. Point S is the reflection of point D across line X. Point A is the reflection of point D across line O. Point I is the reflection of point D across point A. Let U be the reflection of point Y across point I. Point T is the reflection of point U across point A. 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}


[User]
Let NF = 1. Let R be the midpoint of segment FN. Point Y is obtained by rotating point F by angle 7pi/12 about point R. Point Q is the midpoint of segment YR. Point E is the reflection of point F across segment NY. Find the length of segment QE. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JR = 12. Let K be the midpoint of segment RJ. Line A is the perpendicular bisector of segment RK. Point W lies on line A at distance 1 from point R. Line E passes through point K perpendicular to segment RJ. Let C be the reflection of point J across point W. Line V is the reflection of line E across line A. Let G be the reflection of line A across line E. Let X be the angle bisector of angle WRK. Let L be the reflection of line X through point C. Point Y is on line G. Line Z passes through point Y parallel to line L. Point O is the intersection of lines V and Z. Find the distance between points O 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 BE = 11. Let G be the line through points B and E. Let P be the line through point B perpendicular to line G. Point N lies on line P at distance 7 from point E. Let X be the reflection of point N across point B. Point I is the reflection of point E across point X. Let W be the midpoint of segment IB. Point Q is the reflection of point W across segment XI. What is the length of segment QW? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let EY = 7. Let S be the midpoint of segment EY. Let K be the vector from point S to point Y. Line X is the perpendicular bisector of segment YS. Point H is the translation of point Y by vector K. Point Z is on line X with |ZH| = 9. Let L be the midpoint of segment SZ. Let F be the midpoint of segment HZ. Let U be the midpoint of segment HL. Find the distance between points F and U. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SB = 9. Point W is the midpoint of segment SB. Let P be the midpoint of segment BW. Line V is the perpendicular bisector of segment SW. Line Y is the reflection of line V across point W. Let D be the reflection of point S across point P. Point I lies on line Y at distance 9 from point W. Let M be the line through point S parallel to segment DI. Let A be the midpoint of segment DI. Point L is the reflection of point A across line M. What is the distance between points W and L? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let QC = 9. Let I be the reflection of point C across point Q. Let B be the rotation of point Q by angle 5pi/6 around point I. Let G be the midpoint of segment BI. Let H be the reflection of point I across point B. Let S be the angle bisector of angle BQC. Point R is obtained by rotating point H by angle pi/4 about point C. Point U is the reflection of point R across segment BI. Let D be the midpoint of segment GU. Point Y is the reflection of point U across line S. Let M be the midpoint of segment YG. Find 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}


[User]
JH = 10. Let B be the midpoint of segment HJ. Point T is obtained by rotating point H by angle 5pi/6 about point J. Let V be the midpoint of segment TB. Let I be the rotation of point T by angle pi/3 around point J. Point D is the midpoint of segment HV. Let W be the midpoint of segment IJ. Line C is the angle bisector of angle BDW. Let G be the line through point D perpendicular to line C. Point M is the reflection of point W across line G. Find the length of segment MW. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OP = 7. Let A be the vector from point O to point P. Point F is the translation of point P by vector A. Point M is the translation of point F by vector A. Let W be the rotation of point M by angle 2pi/3 around point P. Point L is the midpoint of segment FW. Let S be the midpoint of segment LP. Segments WP and OS intersect at point B. What is 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]
OF = 9. Point H is obtained by rotating point O by angle 5pi/6 about point F. Line A is the angle bisector of angle FHO. Point X is the reflection of point O across line A. Line C passes through point H perpendicular to line A. Point G is the reflection of point X across line C. Let T be the midpoint of segment GX. Point I is the reflection of point H across segment GX. Find 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]
DR = 11. Line F is the perpendicular bisector of segment RD. Let W be the midpoint of segment RD. Line I is the perpendicular bisector of segment WD. Point P is on line I with |PW| = 6. Point Z is the midpoint of segment PD. Point V is the midpoint of segment WZ. Let A be the line through point Z perpendicular to line F. Point X is the reflection of point R across line A. Point C is the midpoint of segment XV. Point K is the reflection of point Z across line F. What is the length of segment CK? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WC = 10. Let D be the reflection of point C across point W. Let A be the perpendicular bisector of segment CD. Point Q is obtained by rotating point W by angle 7pi/12 about point C. Point H is obtained by rotating point D by angle pi/4 about point Q. Point J is obtained by rotating point W by angle 5pi/6 about point C. Let X be the midpoint of segment HC. Line I is the perpendicular bisector of segment JX. Point F is the reflection of point Q across line A. Let T be the reflection of line I across line A. Point G is the intersection of line T and segment FC. Point M is the reflection of point G across line A. Find the length of segment MJ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FH = 3. Line B passes through point F perpendicular to segment HF. Point K is the reflection of point H across line B. Point L is the midpoint of segment HF. Line Q is the perpendicular bisector of segment HF. Let R be the rotation of point F by angle 5pi/12 around point K. Line M passes through point R perpendicular to line Q. Let P be the midpoint of segment LR. Let V be the rotation of point H by angle 11pi/12 around point K. Point Y is the midpoint of segment FR. Segments FR and VP intersect at point W. Line C passes through point Y parallel to line M. Point U is the reflection of point L across line C. Point D is the reflection of point W across point U. 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]
ZY = 7. Point W is obtained by rotating point Y by angle 5pi/12 about point Z. Let T be the angle bisector of angle ZWY. Point L is the reflection of point Z across line T. Point S is the midpoint of segment LZ. Let R be the midpoint of segment WS. Point U is the midpoint of segment RZ. Line M passes through point R perpendicular to segment WU. Point A is the reflection of point S across line M. Find the distance between points A and W. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LZ = 8. Let F be the rotation of point L by angle pi/2 around point Z. Point X is the midpoint of segment FL. Point V is obtained by rotating point F by angle pi/3 about point Z. Let W be the perpendicular bisector of segment XL. Point R is the midpoint of segment ZF. Point D lies on line W at distance 6 from point L. Point T is the reflection of point R across segment VL. Let Q be the reflection of point T across point D. Let M be the midpoint of segment RV. Point I is the reflection of point M across segment XQ. What is the length of segment IX? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RQ = 4. Let X be the rotation of point Q by angle 7pi/12 around point R. Point O is obtained by rotating point R by angle pi/4 about point Q. Point V is the reflection of point O across point X. Point F is obtained by rotating point X by angle pi/12 about point O. Let P be the rotation of point F by angle 11pi/12 around point Q. Point N is the midpoint of segment QO. Let I be the reflection of point F across point X. Let B be the reflection of point P across point F. Let A be the midpoint of segment BI. Segments NF and VO intersect at point W. What is the length of segment WA? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RG = 11. Point U is the reflection of point R across point G. Let P be the rotation of point U by angle 5pi/6 around point G. Point Q is the reflection of point P across segment GR. Let B be the midpoint of segment GQ. Point M is obtained by rotating point B by angle pi/2 about point U. Point H is the reflection of point U across point M. Line A passes through point M parallel to segment GR. Let Y be the line through points G and H. Point D is the intersection of lines A and Y. What is the length of segment DP? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let SI = 12. Point L is obtained by rotating point S by angle pi/12 about point I. Let B be the reflection of point S across point L. Let M be the midpoint of segment BL. Point T is the reflection of point M across segment SI. Find the length of segment TM. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
DV = 1. Line H is the perpendicular bisector of segment VD. Line N passes through point V parallel to line H. Point S is on line N with |SV| = 11. Point Z lies on line H at distance 1 from point D. Point I is the reflection of point Z across point D. Point E is on line N with |ES| = 8. Let Q be the reflection of point E across point Z. Let Y be the reflection of point Q across point I. Line A passes through points E and Z. Point P is the reflection of point Y across segment VD. Point U is the reflection of point P across line A. Find the length of segment UQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let AX = 3. Point F is the reflection of point A across point X. Let Z be the rotation of point X by angle pi/2 around point F. Let M be the vector from point Z to point A. Let D be point X translated by vector M. Let Y be the rotation of point D by angle 7pi/12 around point A. Let N be point Y translated by vector M. Point B is the midpoint of segment NZ. Let J be the perpendicular bisector of segment AZ. Line W passes through point B parallel to line J. Let P be the rotation of point A by angle 2pi/3 around point B. Point E is the reflection of point P across line W. Point T is the reflection of point E across segment YB. Segments PX and AZ intersect at point R. Find the length of segment RT. 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 LI = 9. Point B is obtained by rotating point I by angle 5pi/6 about point L. Let Q be the rotation of point I by angle 2pi/3 around point B. Point V is the midpoint of segment BQ. Point R is the reflection of point V across point Q. Let F be the angle bisector of angle VQI. Line Y is the perpendicular bisector of segment VQ. Let D be the reflection of point R across point B. Let U be the reflection of line F across line Y. Let S be the reflection of point D across point L. Point N is the midpoint of segment SV. Line A is the reflection of line Y across line U. Point H is the reflection of point R across line A. Point C is the reflection of point N across line A. Point T is the reflection of point H across point D. Point W is the reflection of point T across segment DC. What is the length of segment WV? 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 NO = 12. Point C is the reflection of point O across point N. Let L be the midpoint of segment ON. Let D be the rotation of point C by angle 11pi/12 around point O. Let A be the reflection of point L across point O. Let V be the angle bisector of angle DLA. Point S is the reflection of point L across point N. Point H is the reflection of point S across line V. Point I is the reflection of point H across point L. What is the distance between points D and I? 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 LJ = 3. Point G is obtained by rotating point J by angle 11pi/12 about point L. Let U be the reflection of point G across point J. Point X is obtained by rotating point J by angle 7pi/12 about point U. Let F be the vector from point J to point X. Let H be point L translated by vector F. Let V be point G translated by vector F. Point A is obtained by rotating point V by angle pi/12 about point U. Line K passes through points H and V. Point M is the reflection of point X across line K. Point S is the intersection of segments AJ and LX. Point C is the reflection of point M across segment AJ. Find 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}


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Let TP = 10. Let Q be the line through point T perpendicular to segment PT. Let M be the rotation of point T by angle pi/4 around point P. Let V be the rotation of point P by angle pi/4 around point T. Let C be the midpoint of segment PT. Point D is the reflection of point M across line Q. Let I be the line through point C perpendicular to segment VM. Let X be the reflection of point M across point D. Point H is the reflection of point X across line I. Find the length of segment XH. 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 OB = 4. Line X is the perpendicular bisector of segment OB. Line G passes through point O perpendicular to line X. Point Z lies on line X. Point R is the reflection of point Z across point O. Point D is the reflection of point R across line G. What is the distance between points D and Z? 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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JI = 2. Line R passes through points J and I. Let P be the reflection of point I across point J. Let H be the rotation of point I by angle 11pi/12 around point J. Point Z is obtained by rotating point I by angle pi/6 about point J. Point S is obtained by rotating point H by angle pi/3 about point I. Line D is the angle bisector of angle PJS. Let O be the reflection of line R across line D. Let Q be the line through point P parallel to line O. Point C is the reflection of point I across line Q. Find the length of segment ZC. 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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RS = 8. Let Y be the midpoint of segment SR. Let L be the rotation of point Y by angle pi/6 around point S. Point N is the reflection of point Y across point R. Point V is the midpoint of segment LN. Point Q is the reflection of point S across point L. Point Z is the midpoint of segment VQ. Find the length of segment ZR. 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 DS = 3. Let W be the rotation of point S by angle 2pi/3 around point D. Point N is the midpoint of segment DS. Let H be the angle bisector of angle WNS. Point P is the reflection of point N across point S. Point J is the reflection of point P across line H. Point R is the reflection of point S across point P. What is the distance between points J and R? 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 WM = 11. Point Y is the midpoint of segment WM. Let P be the perpendicular bisector of segment MW. Point B is the midpoint of segment WY. Point Q is on line P. Point R is the midpoint of segment BQ. Point N is the midpoint of segment QR. Let F be the rotation of point N by angle pi/3 around point W. Point E is the reflection of point F across segment RW. Let K be the line through points E and W. Point H is the reflection of point Y across line K. Point I is the midpoint of segment WH. Find the length of segment IH. Express your answer as a decimal number rounded to 2 decimal places.

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SM = 9. Point J is the midpoint of segment SM. Line E is the perpendicular bisector of segment SM. Let D be the line through point M parallel to line E. Point V is the reflection of point J across line D. Find the distance between points J 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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AN = 1. Let I be the rotation of point N by angle 7pi/12 around point A. Let X be the angle bisector of angle NAI. Point S is the reflection of point I across point N. Line J passes through points S and A. Let K be the rotation of point A by angle pi around point N. Line R is the reflection of line X across line J. Point W is obtained by rotating point S by angle pi/4 about point I. Point Y is the midpoint of segment IN. Let H be the reflection of line X through point I. Lines H and R intersect at point B. Point U is the reflection of point K across segment YB. Find the distance between points W 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 FG = 10. Line A is the perpendicular bisector of segment GF. Let P be the vector from point F to point G. Point U is the midpoint of segment GF. Let E be point U translated by vector P. Point D is the reflection of point E across line A. Line Q is the reflection of line A across point F. Point W is obtained by rotating point D by angle 5pi/6 about point F. Point X is the reflection of point G across line Q. Point N is the reflection of point D across point W. Let V be the reflection of line Q through point E. Point K is the reflection of point N across line V. Let T be the reflection of point D across point K. Point I is the midpoint of segment TG. Line S passes through point K perpendicular to segment EI. Point R is the reflection of point N across line S. What is the length of segment XR? 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 OV = 6. Point L is obtained by rotating point O by angle pi/6 about point V. Let G be the reflection of point V across point O. Let T be the midpoint of segment LG. Line S is the angle bisector of angle TGL. Point J is the reflection of point V across line S. Point R is the reflection of point L across segment OV. What is the distance between points J and R? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YB = 10. Point M is the midpoint of segment YB. Let K be the reflection of point M across point Y. Line R is the perpendicular bisector of segment YM. Line E is the reflection of line R across point M. Point H is the reflection of point K across line E. 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 GC = 12. Line E is the perpendicular bisector of segment GC. Point I is the reflection of point G across point C. Point Q is the reflection of point C across point I. Let B be the reflection of point G across point Q. Point Y is the reflection of point B across line E. Let O be the rotation of point G by angle pi/12 around point Y. Let X be the midpoint of segment OG. Let K be the reflection of point X across point B. Point A is the midpoint of segment BX. Let F be the midpoint of segment KQ. What is the distance between points A 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 VK = 1. Let G be the perpendicular bisector of segment KV. Let Z be the line through point V perpendicular to line G. Let J be the rotation of point K by angle pi/4 around point V. Point Y is the reflection of point K across point J. Line A is the angle bisector of angle YVJ. Point C is the reflection of point Y across line A. Let Q be the reflection of point V across point J. Let N be the reflection of point C across point K. Let F be the reflection of point J across point Q. Point P is the reflection of point N across line Z. Point I is the reflection of point K across point F. Point S is the reflection of point I across line Z. Find the distance between points S 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 MA = 2. Point D is the midpoint of segment MA. Let L be the rotation of point M by angle 5pi/12 around point A. Line O passes through points A and L. Let H be the rotation of point M by angle 5pi/12 around point D. Point E is the reflection of point M across line O. Point Q is the reflection of point L across segment EH. Point Z is the midpoint of segment EH. What is the length of segment ZQ? 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 DV = 12. Let K be the reflection of point V across point D. Line W is the angle bisector of angle VDK. Point E lies on line W at distance 8 from point D. Point L is the reflection of point E across segment DV. Point B is the reflection of point D across point K. Point Z is the midpoint of segment BK. Let O be the midpoint of segment BZ. Find the distance between points L and O. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VB = 2. Point Z is obtained by rotating point B by angle pi/2 about point V. Let T be the reflection of point V across point Z. Point E is the reflection of point B across point Z. Point M is obtained by rotating point E by angle pi/3 about point B. Point H is the reflection of point T across segment ZM. Point Y is the midpoint of segment HB. Point G is the reflection of point Z across segment VB. Find the distance between points Y and G. 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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EM = 7. Let G be the rotation of point M by angle pi/3 around point E. Let L be the reflection of point M across point E. Point F is the midpoint of segment MG. Let I be the midpoint of segment LG. Point W is the midpoint of segment ME. Point H is the midpoint of segment GI. Point C is the reflection of point W across point G. What is the measure of angle CFH, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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NV = 8. Point Y is obtained by rotating point N by angle pi/3 about point V. Point E is the reflection of point Y across point N. Let T be the midpoint of segment EV. Let I be the midpoint of segment VN. Let B be the reflection of point N across point T. What is the length of segment BI? 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 AW = 8. Point C is obtained by rotating point W by angle pi/6 about point A. Point V is the midpoint of segment WC. Let J be the reflection of point W across point C. Let M be the midpoint of segment AJ. Let G be the rotation of point W by angle 7pi/12 around point A. Line E is the perpendicular bisector of segment GC. Line F passes through point V parallel to line E. Line B passes through points W and M. Point P is the intersection of lines F and B. Find the distance between points C 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 GB = 8. Let N be the vector from point B to point G. Point T is the midpoint of segment GB. Point H is the reflection of point T across point G. Let L be the rotation of point B by angle pi/2 around point T. Point Q is the reflection of point G across point L. Let Y be point H translated by vector N. Let J be the midpoint of segment LY. Point M is the midpoint of segment YQ. Let E be the midpoint of segment JM. Find the length of segment HE. Express your answer as a decimal number rounded to 2 decimal places.

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YR = 5. Let W be the perpendicular bisector of segment RY. Let Z be the reflection of point R across point Y. Point N lies on line W at distance 9 from point Y. Point B is the reflection of point Y across point N. Let S be the midpoint of segment ZN. Line P is the perpendicular bisector of segment SB. Point X is the reflection of point R across line P. Let F be the midpoint of segment BX. Find the length of segment XF. 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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LG = 8. Let T be the vector from point G to point L. Point E is the midpoint of segment GL. Let N be point E translated by vector T. Point M is on segment GL. Line R passes through point G perpendicular to segment GL. Point I is the midpoint of segment NM. Point X is the reflection of point I across line R. Point S is the reflection of point I across point L. Find the distance between points X and S. 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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HS = 1. Let X be the midpoint of segment SH. Let L be the rotation of point H by angle 11pi/12 around point S. Let N be the line through point S perpendicular to segment SH. Let D be the reflection of point L across point X. Let W be the midpoint of segment DS. Point J is the reflection of point L across line N. What is the distance between points W 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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Let MQ = 6. Point R is obtained by rotating point M by angle 7pi/12 about point Q. Let A be the angle bisector of angle QRM. Let P be the reflection of point M across point R. Let Z be the reflection of line A through point P. Let C be the midpoint of segment QR. Point X is the reflection of point C across line Z. What is the length of segment XR? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OV = 3. Let U be the rotation of point O by angle pi/3 around point V. Point H is obtained by rotating point V by angle 5pi/6 about point U. Let E be the perpendicular bisector of segment HV. Let L be the midpoint of segment VU. Point J is the reflection of point L across line E. Point Z is the midpoint of segment LJ. Point I is the midpoint of segment OU. Let G be the rotation of point U by angle 11pi/12 around point Z. Let M be the reflection of point H across point L. Line B is the angle bisector of angle GZH. Point A is the midpoint of segment MO. Let W be the line through point L parallel to line B. Line W and segment VI intersect at point N. What is the distance between points N and A? 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 FT = 4. Let B be the rotation of point F by angle pi/6 around point T. Let X be the reflection of point B across point F. Let S be the rotation of point F by angle 3pi/4 around point X. Let R be the midpoint of segment FB. Let W be the rotation of point X by angle pi around point R. Point V is the reflection of point X across segment TF. Let N be the rotation of point B by angle 5pi/12 around point T. Point M is the reflection of point W across segment TF. Point C is the midpoint of segment MT. Let P be the rotation of point N by angle pi/2 around point S. Let K be the midpoint of segment BV. Let H be the reflection of point S across point P. Segments CK and TF intersect at point J. Find the length of segment JH. Express your answer as a decimal number rounded to 2 decimal places.

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JX = 11. Point T is the reflection of point X across point J. Line A passes through points J and X. Line Q passes through point X perpendicular to line A. Let V be the rotation of point T by angle pi/3 around point J. Point U is the reflection of point V across line Q. Let F be the reflection of point T across point X. Point K is the midpoint of segment UJ. Point Z is the reflection of point F across point U. What is the length of segment KZ? 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 HM = 11. Let F be the rotation of point H by angle pi/12 around point M. Let B be the vector from point F to point M. Point X is the translation of point M by vector B. Point U is obtained by rotating point H by angle 7pi/12 about point X. Point G is the reflection of point H across point M. Let Y be the reflection of point U across point G. Let S be the line through points X and Y. Point I is the reflection of point Y across point H. Point P is the reflection of point I across line S. What is the length of segment FP? Express your answer as a decimal number rounded to 2 decimal places.

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ET = 4. Let A be the rotation of point E by angle pi/6 around point T. Point K is obtained by rotating point E by angle pi/12 about point A. Let C be the line through point E perpendicular to segment TE. Point S is the reflection of point A across line C. Let J be the reflection of point A across point K. Let Y be the midpoint of segment JS. Let D be the line through point E parallel to segment JS. Point L is the midpoint of segment TY. Point X is the reflection of point Y across line D. Find the distance between points X and L. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ZJ = 7. Let U be the rotation of point Z by angle pi/4 around point J. Line C is the perpendicular bisector of segment JZ. Point W is the reflection of point U across line C. Let L be the line through point U parallel to line C. Line D is the reflection of line C across point U. Let F be the reflection of point U across point W. Line M is the perpendicular bisector of segment JF. Segment ZJ and line D intersect at point X. Segment ZJ and line L intersect at point K. Let S be the line through point F perpendicular to segment ZJ. Line R is the reflection of line M across line S. Point N is obtained by rotating point J by angle 2pi/3 about point U. Let E be the line through point N parallel to segment ZJ. Let T be the reflection of point K across point Z. Let H be the angle bisector of angle UXT. Point I is the intersection of line R and segment FJ. Point G is the reflection of point I across line H. Segment WX and line E intersect at point Y. Point B is the midpoint of segment YF. Let V be the line through points B and X. Point P is the reflection of point G across line V. What is the distance between points Y and P? 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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WT = 9. Point L is obtained by rotating point T by angle 7pi/12 about point W. Let P be the perpendicular bisector of segment TW. Point Z is the reflection of point L across point T. Let Q be the line through point L parallel to line P. Point H is the reflection of point T across point Z. Let K be the line through points H and Z. Point Y is the reflection of point W across line K. Point N is the reflection of point H across line Q. What is the length of segment YN? 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 RO = 8. Let W be the rotation of point O by angle 5pi/12 around point R. Let C be the vector from point R to point O. Let A be point O translated by vector C. Line E passes through points A and R. Let U be the midpoint of segment WO. Line V passes through point A parallel to segment OW. Line X is the reflection of line E across line V. Point N is the reflection of point U across line X. What is the length of segment AN? Express your answer as a decimal number rounded to 2 decimal places.

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LQ = 10. Point R is the midpoint of segment QL. Let P be the rotation of point R by angle 5pi/6 around point L. Point S is the reflection of point P across point R. Point Z is the reflection of point P across point L. Line E passes through points S and P. Let V be the line through point Z parallel to line E. Let Y be the reflection of line V across line E. Point N is the reflection of point S across line Y. What is the length of segment ZN? 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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UD = 6. Point B is obtained by rotating point D by angle pi/3 about point U. Point T is the midpoint of segment UD. Point J is obtained by rotating point T by angle 7pi/12 about point D. Let L be the midpoint of segment DJ. Point A is the midpoint of segment BU. Line Q passes through point D parallel to segment BU. Point X is the reflection of point A across point L. Point H is the reflection of point J across line Q. What is the distance between points X and H? 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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KW = 7. Let C be the rotation of point W by angle 5pi/6 around point K. Point N is the midpoint of segment WK. Let V be the midpoint of segment CN. Point M is the reflection of point V across segment WK. Let E be the reflection of point M across point V. Find 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}


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UK = 4. Point L is the midpoint of segment UK. Let Q be the line through point K perpendicular to segment KL. Point H is the reflection of point U across line Q. Point V is the midpoint of segment KL. What is the distance between points V and H? Express your answer as a decimal number rounded to 2 decimal places.

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SB = 6. Point V is the midpoint of segment SB. Let W be the perpendicular bisector of segment VB. Let E be the rotation of point S by angle pi/2 around point V. Let N be the angle bisector of angle EVB. Point F is the midpoint of segment SV. Point H is on line W with |HS| = 6. Line K is the angle bisector of angle HFS. Let A be the midpoint of segment EV. Let R be the reflection of line K across line N. Line D is the perpendicular bisector of segment VE. Point Y is the reflection of point V across line R. Line I is the reflection of line D across point Y. Point U is the reflection of point F across line I. Find the length of segment UA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let JN = 7. Line Y passes through point N perpendicular to segment JN. Point E is the midpoint of segment JN. Point H is the midpoint of segment EN. Let X be the reflection of point E across point J. Point P is the midpoint of segment XJ. Point C is the midpoint of segment PE. Let W be the line through point C parallel to line Y. Point U lies on line W. Point Z is the reflection of point P across point H. Let A be the angle bisector of angle UZX. Point T is the reflection of point C across line A. What is the length of segment TZ? 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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CJ = 1. Let O be the reflection of point C across point J. Point W is the reflection of point O across point C. Point F is the midpoint of segment CJ. Let L be the rotation of point O by angle 5pi/12 around point W. Let G be the midpoint of segment CF. Point Z is the reflection of point F across segment LG. Point N is the reflection of point Z across segment CJ. Find the length of segment ZN. 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 VS = 6. Let J be the rotation of point V by angle pi/6 around point S. Line H is the perpendicular bisector of segment SJ. Point U is the reflection of point V across line H. Let W be the rotation of point U by angle 3pi/4 around point J. Let Q be the rotation of point W by angle 7pi/12 around point V. Let Y be the midpoint of segment JU. Let D be the reflection of line H through point W. Point A is the reflection of point Q across segment SJ. Point G is the reflection of point A across line D. Find the distance between points G and Y. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QK = 4. Line I is the perpendicular bisector of segment QK. Point R is the midpoint of segment QK. Let Z be the rotation of point K by angle 5pi/6 around point Q. Let S be the reflection of point R across point Z. Point Y is the reflection of point Z across line I. Point U is the reflection of point Y across segment SQ. Find the length of segment RU. Express your answer as a decimal number rounded to 2 decimal places.

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Let SV = 2. Let G be the reflection of point S across point V. Point W is the midpoint of segment GV. Let N be the perpendicular bisector of segment SG. Point R is the reflection of point W across line N. Find the distance between points G and R. 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 TG = 7. Point X is the midpoint of segment GT. Let N be the rotation of point G by angle pi/12 around point X. Point M is the reflection of point N across point X. Let O be the line through points N and X. Line V passes through point T perpendicular to line O. Let K be the reflection of point T across point M. Point Y is the reflection of point M across line V. What is the distance between points K 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 JN = 4. Point W is obtained by rotating point N by angle pi/2 about point J. Point C is the reflection of point N across point W. Let O be the midpoint of segment JN. Point R is the midpoint of segment OC. Let Z be the perpendicular bisector of segment RN. Let M be the line through point W parallel to line Z. Line M and segment OC intersect at point Q. Point I is the midpoint of segment QC. Point F is the reflection of point N across segment OW. What is the length of segment FI? Express your answer as a decimal number rounded to 2 decimal places.

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Let FT = 11. Let U be the perpendicular bisector of segment TF. Let G be the rotation of point F by angle 5pi/12 around point T. Let M be the midpoint of segment GF. Let L be the reflection of line U through point T. Point J is the reflection of point G across line U. Let P be the midpoint of segment TG. Let E be the line through point P perpendicular to line L. Let H be the line through point G parallel to line E. Let B be the perpendicular bisector of segment GF. Let W be the rotation of point J by angle 11pi/12 around point G. Let V be the reflection of line B across line H. Point I is the reflection of point T across line V. Let Y be the reflection of point M across point W. Point X is the reflection of point Y across point I. Find the distance between points X and P. 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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SK = 5. Let L be the perpendicular bisector of segment SK. Point J is on line L with |JK| = 9. Point P is the reflection of point J across segment SK. Point F is the midpoint of segment SK. Let E be the reflection of point P across point J. Let T be the midpoint of segment FK. Let I be the line through points S and E. Point C is the reflection of point P across point K. Point Y is the reflection of point T across line I. Find 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}


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Let WL = 2. Point U is obtained by rotating point W by angle 3pi/4 about point L. Point E is obtained by rotating point U by angle pi/12 about point L. Point I is the midpoint of segment WL. Point Q is the midpoint of segment LE. Line F passes through point L perpendicular to segment WL. Point R is the reflection of point I across line F. What is the length of segment QR? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IW = 3. Let K be the rotation of point W by angle pi/4 around point I. Point C is obtained by rotating point K by angle 5pi/12 about point W. Point B is the reflection of point W across segment IK. Point J is the midpoint of segment KC. Let O be the reflection of point B across point C. Let S be the reflection of point O across point J. Find the length of segment OS. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FD = 12. Let H be the midpoint of segment FD. Point P is the midpoint of segment FH. Let I be the rotation of point D by angle 2pi/3 around point P. Let L be the line through point I parallel to segment DF. Point Z is the midpoint of segment IH. Point J is the reflection of point Z across line L. Point R is the reflection of point D across segment JI. Let E be the midpoint of segment ZD. What is the distance between points R and E? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FA = 9. Let H be the rotation of point F by angle 5pi/12 around point A. Point Y is the midpoint of segment AH. Point D is the midpoint of segment AY. Let N be the reflection of point F across point D. Point I is the midpoint of segment AN. Point K is the reflection of point I across segment ND. What is the length of segment NK? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HW = 3. Point S is obtained by rotating point H by angle 2pi/3 about point W. Let E be the perpendicular bisector of segment WS. Let U be the rotation of point S by angle 11pi/12 around point W. Let D be the reflection of point U across point W. Let X be the reflection of line E through point U. Point G is the midpoint of segment DH. Point L is the reflection of point D across line X. Find the length of segment GL. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LZ = 11. Let O be the rotation of point L by angle 5pi/6 around point Z. Let T be the rotation of point O by angle 5pi/6 around point Z. Point U is the midpoint of segment OZ. Let G be the line through point T perpendicular to segment OZ. Let S be the midpoint of segment TO. Line K passes through point S parallel to line G. Point R is the midpoint of segment US. Point D is the reflection of point R across line K. Find the distance between points D 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 DY = 9. Point L is obtained by rotating point Y by angle 7pi/12 about point D. Let R be the midpoint of segment YD. Let G be the perpendicular bisector of segment RL. Point F is the reflection of point Y across line G. Point E is the reflection of point F across segment YD. Point M is the reflection of point F across point E. Find 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}


[User]
PM = 9. Point B is the midpoint of segment PM. Line T is the perpendicular bisector of segment MB. Line O passes through point P parallel to line T. Point A is obtained by rotating point B by angle pi/6 about point M. Point Q is the midpoint of segment PA. Point G is the midpoint of segment AB. Point K is the reflection of point Q across line O. Line W is the perpendicular bisector of segment AP. Let Y be the line through point K parallel to line W. Point U is the reflection of point G across line O. Point D is the reflection of point U across segment AP. Point N is the reflection of point D across line Y. What is the distance between points N 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 IQ = 7. Point O is the midpoint of segment QI. Let R be the midpoint of segment OQ. Let D be the reflection of point R across point O. Let X be the line through point Q perpendicular to segment QR. Point B is the reflection of point R across line X. Let F be the midpoint of segment QB. Find the distance between points F and D. Express your answer as a decimal number rounded to 2 decimal places.

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