[User]
Let FS = 8. Let X be the rotation of point F by angle pi/12 around point S. Point A is the midpoint of segment SX. Let Y be the line through points F and S. Let FE = 2. Line G is the reflection of line Y across point E. Point K is the reflection of point X across line G. Line Q is the angle bisector of angle FEA. Point Z lies on line Q. Let T be the line through point K perpendicular to line G. Point H is the reflection of point F across line T. Point R is the reflection of point H across segment ZF. What is the length of segment RF? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let KF = 2. Let N be the midpoint of segment KF. Z is placed 11 units away from N. Line I is the perpendicular bisector of segment ZN. Point E is on line I. Point X is the reflection of point E across segment FZ. Let U be the line through points X and K. Point A is the reflection of point N across line U. What is the length of segment AK? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HZ = 9. Let Q be the line through points Z and H. Point P lies on line Q. Point Y is the reflection of point Z across point H. Line F passes through point Z perpendicular to line Q. Let X be the midpoint of segment ZP. Point G is the reflection of point Y across line F. Line B is the perpendicular bisector of segment GX. Point K is the reflection of point Z across line B. Find the distance between points X and K. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VF = 10. Let H be the perpendicular bisector of segment VF. Let B be the rotation of point F by angle pi/4 around point V. Line A passes through points B and F. Line I passes through point V parallel to line A. Line P is the reflection of line H across point B. Let Y be the reflection of line P across line I. Line X is the reflection of line H across line Y. Point T is the midpoint of segment VF. Point J is the intersection of lines Y and X. Find the distance between points T 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 IT = 9. Point X is the midpoint of segment TI. Point H is obtained by rotating point I by angle pi/4 about point T. Let W be the reflection of point H across point X. Line S is the perpendicular bisector of segment TI. Point E is the reflection of point W across line S. Line D is the reflection of line S across point W. Point J is obtained by rotating point E by angle pi/12 about point I. Line N passes through points J and T. Let Y be the reflection of line S through point E. Point C is the reflection of point I across line N. Let L be the line through points C and I. Point M is the intersection of lines Y and L. Point U is the intersection of line D and segment TI. What is the length of segment MU? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WB = 5. Let G be the reflection of point W across point B. Line I is the perpendicular bisector of segment WG. Point V lies on line I at distance 9 from point W. Let H be the midpoint of segment VB. Point O is the midpoint of segment HB. Point T is the midpoint of segment HG. Point A is the reflection of point T across line I. What is the distance between points O and A? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let IJ = 7. Point H is obtained by rotating point I by angle pi/6 about point J. Line L is the perpendicular bisector of segment JI. Line T is the angle bisector of angle HJI. Let V be the line through point H parallel to line T. Let B be the reflection of line V across line T. Line P is the reflection of line T across line L. Let W be the midpoint of segment HI. Let E be the midpoint of segment JW. Point U is the reflection of point H across line B. Line M is the reflection of line V across point I. Point F is the midpoint of segment UJ. Point S is the intersection of lines P and M. Point Z is the reflection of point E across point F. Find the distance between points Z 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 OH = 10. Let I be the midpoint of segment HO. Let C be the rotation of point I by angle 2pi/3 around point H. Point K is the midpoint of segment IC. Let U be the line through points C and H. Let F be the line through point C perpendicular to line U. Point Y is the reflection of point H across line F. What is the length of segment KY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GB = 11. Let C be the line through points B and G. Let F be the rotation of point B by angle pi/3 around point G. Point I lies on line C. V is placed 10 units away from I. Let U be the perpendicular bisector of segment GI. Let J be the midpoint of segment FB. Point Q is the midpoint of segment JF. Point T is the reflection of point V across line U. Point D is the midpoint of segment QT. Let A be the midpoint of segment TG. Find the distance between points A and D. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AL = 8. Let J be the midpoint of segment LA. Point P is the reflection of point J across point A. Point S is obtained by rotating point A by angle 3pi/4 about point P. Let W be the midpoint of segment LS. Line V is the perpendicular bisector of segment AP. Point N is the reflection of point L across point A. Point T is the reflection of point L across line V. Point C is the reflection of point N across point T. What is the distance between points W and C? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
UH = 9. Let A be the perpendicular bisector of segment UH. Line T passes through point H parallel to line A. Point P lies on line T. Let M be the midpoint of segment HP. Point C is the midpoint of segment UP. What is the distance between points M 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 GT = 8. Point U is obtained by rotating point T by angle 5pi/12 about point G. Point M is the midpoint of segment TG. Let W be the line through points U and G. Let J be the reflection of line W through point T. Point X is the reflection of point T across point U. Point Y is the reflection of point G across line J. Line C is the reflection of line W across point X. Line V passes through point M parallel to line C. Let L be the reflection of line C across line V. Line L and segment YM intersect at point O. What is 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 GW = 8. Point A is obtained by rotating point W by angle pi/6 about point G. Let S be the midpoint of segment AW. Point P is the midpoint of segment AS. Line V passes through point P perpendicular to segment WG. Point E is the reflection of point G across line V. Point O is the reflection of point E across point P. Find the length of segment SO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let YJ = 12. Point A is obtained by rotating point J by angle 5pi/12 about point Y. Point Q is obtained by rotating point Y by angle pi/3 about point J. Let C be the vector from point J to point Y. Let U be the rotation of point Q by angle 7pi/12 around point Y. Let M be the perpendicular bisector of segment YU. Let K be point U translated by vector C. Let Z be the line through point K perpendicular to line M. Point W is the midpoint of segment AK. Let X be the line through point W parallel to line Z. Line N is the reflection of line Z across point W. Point F is the reflection of point J across line Z. Line O passes through points F and A. Let R be the reflection of line X across line N. Point I is the reflection of point W across line R. Point E is the reflection of point I across line O. What is the distance between points E and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GW = 9. Let H be the rotation of point G by angle pi/4 around point W. Point L is the midpoint of segment WG. Point C is the midpoint of segment HL. Point J is the reflection of point C across segment GW. What is the distance between points L and J? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YM = 1. Let L be the reflection of point M across point Y. Line G is the perpendicular bisector of segment LM. Let W be the midpoint of segment YL. Line O is the reflection of line G across point W. Point Z is the reflection of point Y across line O. Find the distance between points Z and Y. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SP = 7. Let M be the rotation of point P by angle pi/12 around point S. Point D is obtained by rotating point M by angle pi/6 about point P. Let J be the perpendicular bisector of segment PS. Point F is obtained by rotating point M by angle pi about point S. Let C be the line through points F and S. Line E is the reflection of line C across line J. Point R is the reflection of point D across line E. Point Y is the midpoint of segment FR. Point A is the reflection of point D across point M. What is the length of segment AY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AN = 4. Let W be the vector from point A to point N. Let K be the midpoint of segment NA. Let H be point K translated by vector W. Point B is obtained by rotating point K by angle 5pi/12 about point N. Let I be the midpoint of segment NK. Let Z be the line through point A perpendicular to segment HN. Point F is the reflection of point A across point B. Let J be the rotation of point N by angle pi/2 around point B. Let P be the line through point I parallel to line Z. Point S is the midpoint of segment FI. Point C is the reflection of point S across line P. Point X is the reflection of point A across point H. Point O is the reflection of point X across point J. What is the length of segment OC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let EA = 11. Point O is obtained by rotating point A by angle 5pi/12 about point E. Point C is the reflection of point E across point O. Let H be the reflection of point O across point C. Let N be the perpendicular bisector of segment AE. Point U is the reflection of point E across point H. Line I passes through point C perpendicular to line N. Point X is the reflection of point C across point E. Point D is the reflection of point U across line I. What is the distance between points D and X? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let YB = 11. Point D is obtained by rotating point B by angle 7pi/12 about point Y. Let N be the perpendicular bisector of segment YB. Let J be the reflection of line N through point D. Let H be the reflection of line J through point Y. Point R is the reflection of point D across line H. Point G is the reflection of point R across segment DB. 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}


[User]
MZ = 6. Point Q is the midpoint of segment MZ. Let G be the vector from point Z to point Q. Let N be the perpendicular bisector of segment ZQ. Let L be the line through points Z and Q. Point C is obtained by rotating point Q by angle pi/4 about point M. Let A be the perpendicular bisector of segment CZ. Line H is the reflection of line A across line L. Let V be point C translated by vector G. Line J is the reflection of line A across point C. Line T is the reflection of line N across point V. Line D is the reflection of line L across line H. Point R is the intersection of lines T and D. Point S is the intersection of segment ZV and line J. Find the length of segment RS. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
DR = 6. Let C be the reflection of point R across point D. Let I be the line through points C and R. Line B passes through point D perpendicular to line I. Let M be the rotation of point D by angle 2pi/3 around point C. Let W be the line through point C perpendicular to line I. Point O is the reflection of point R across line W. Point J is the reflection of point M across point R. Let F be the reflection of point D across point J. Point Y is the midpoint of segment RF. Point Z is obtained by rotating point D by angle 3pi/4 about point C. Point X is the reflection of point Z across line B. Let A be the reflection of point X across point C. Point N is the midpoint of segment OJ. Let K be the reflection of point A across point N. What is the length of segment KY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PU = 6. Point A is obtained by rotating point P by angle pi/6 about point U. Point V is the reflection of point P across point U. Point Y is obtained by rotating point V by angle pi/12 about point P. Point Z is the reflection of point V across point P. Let M be the reflection of point V across point Z. Point G is the midpoint of segment VU. Let R be the line through point M perpendicular to segment VP. Let D be the reflection of point P across point Z. Point Q is the reflection of point G across segment AZ. Line B passes through point P parallel to line R. Point T is the reflection of point D across segment YP. Point W is the reflection of point M across line B. Point S is the midpoint of segment QT. What is the length of segment WS? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
DW = 2. Let E be the midpoint of segment WD. Point K is the midpoint of segment EW. Let M be the rotation of point W by angle 3pi/4 around point D. Point H is the reflection of point K across point E. Let V be the reflection of point M across point D. Point T is the midpoint of segment VK. What is the distance between points T 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 MO = 10. Line S is the perpendicular bisector of segment OM. Let N be the rotation of point O by angle 3pi/4 around point M. Let T be the midpoint of segment MN. Point U is obtained by rotating point N by angle 5pi/12 about point T. Let R be the midpoint of segment UM. Point P is the midpoint of segment UT. Point H is the midpoint of segment NP. Let F be the reflection of line S through point H. Let Z be the perpendicular bisector of segment UH. Point L is the reflection of point R across line F. Let C be the reflection of line Z through point N. Point I is the reflection of point L across line C. Let V be the midpoint of segment MI. What is the length of segment UV? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let AG = 4. Point H is the midpoint of segment GA. Let R be the vector from point G to point A. Let F be point A translated by vector R. Point N is obtained by rotating point F by angle pi/4 about point H. Point Q is the translation of point F by vector R. Line O is the perpendicular bisector of segment NH. Point T is the reflection of point G across line O. Let K be the rotation of point A by angle pi around point T. Line L is the angle bisector of angle AFK. Let Y be the line through point K parallel to line L. Let S be the midpoint of segment KN. Point P is the reflection of point Q across line Y. 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]
QA = 5. Point M is the reflection of point Q across point A. Line B passes through point Q perpendicular to segment QA. Line D is the reflection of line B across point M. Line X is the reflection of line D across line B. Let V be the perpendicular bisector of segment QA. Point I lies on line V. Point T is the reflection of point I across line X. What is the distance between points T 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 CD = 5. Point W is the midpoint of segment DC. Point K is obtained by rotating point C by angle pi/4 about point D. Point X is obtained by rotating point D by angle pi/2 about point W. Let H be the midpoint of segment XC. Let M be the midpoint of segment KW. Point T is the midpoint of segment HX. Let B be the reflection of point H across point M. What is the length of segment TB? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
MY = 10. Point G is obtained by rotating point Y by angle 11pi/12 about point M. Line T is the perpendicular bisector of segment MG. Let F be the line through point Y parallel to line T. Point Q is the reflection of point G across line F. Line D passes through point Q parallel to line F. Line J is the reflection of line F across line D. Point N is the midpoint of segment QY. Point R is the reflection of point N across line J. Point A is the reflection of point R across segment MY. What is the length of segment AM? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VJ = 9. Point A is the midpoint of segment VJ. Let C be the reflection of point J across point V. Let R be the rotation of point C by angle pi/2 around point A. Line Y is the angle bisector of angle RCJ. Point D is the reflection of point C across point R. Line L is the perpendicular bisector of segment CD. Point Q is the midpoint of segment RV. Line T is the reflection of line Y across line L. Point P is obtained by rotating point A by angle pi/2 about point V. Point X is obtained by rotating point A by angle 5pi/12 about point J. Point M is the midpoint of segment DX. Point H is the reflection of point M across line T. Let G be the reflection of point J across point Q. Point I is the reflection of point H across segment PA. Let N be the reflection of point I across point V. 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]
RD = 5. Let U be the vector from point R to point D. Point Y is obtained by rotating point D by angle 7pi/12 about point R. Let O be the midpoint of segment RY. Let X be point O translated by vector U. Point A is the translation of point Y by vector U. Point J is the midpoint of segment DX. Let H be the reflection of point J across point A. Let G be point X translated by vector U. Let Z be the reflection of point Y across point X. Point I is the reflection of point R across point Z. Let W be the angle bisector of angle GIX. Point E is the reflection of point J across line W. Line N passes through points Y and Z. Line M is the reflection of line W across point H. Let S be the line through point O perpendicular to line N. Point V is the midpoint of segment EO. Let B be the reflection of line S through point V. Lines B and M intersect at point L. What is the distance between points L and H? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
CP = 6. Point A is obtained by rotating point P by angle pi/2 about point C. Let D be the rotation of point A by angle pi/4 around point P. Let Z be the perpendicular bisector of segment CP. Let H be the reflection of point A across point D. Line W is the reflection of line Z across point P. Point B is the reflection of point H across line W. Point V is the reflection of point B across segment HC. What is the distance between points B and V? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
TB = 7. Line C passes through points B and T. Point R is the midpoint of segment TB. Let W be the line through point T perpendicular to line C. Point J is on line W with |JT| = 6. Point S is the reflection of point J across line C. Point X is obtained by rotating point R by angle pi/6 about point S. Point Q is the reflection of point T across segment RX. Point O is the reflection of point J across segment RX. Let E be the midpoint of segment SO. What is the distance between points E 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 RY = 8. Point A is obtained by rotating point R by angle pi/3 about point Y. Let D be the rotation of point A by angle 7pi/12 around point R. Point X is obtained by rotating point A by angle pi/6 about point R. Line Q passes through points Y and D. Point P is the reflection of point A across line Q. Let U be the line through point Y perpendicular to line Q. Point W is the reflection of point X across line U. Point F is the reflection of point W across line Q. Find the length of segment FP. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VZ = 10. Let T be the rotation of point V by angle pi/6 around point Z. Let E be the rotation of point T by angle 7pi/12 around point Z. Let Q be the angle bisector of angle ETZ. Point X is the midpoint of segment EZ. Let K be the midpoint of segment TE. Point O is the reflection of point X across line Q. What is the length of segment KO? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YJ = 2. Point U is the midpoint of segment JY. Point M is the reflection of point Y across point J. Point Q is obtained by rotating point M by angle pi/6 about point Y. Point W is obtained by rotating point Q by angle pi/3 about point M. Let C be the rotation of point J by angle pi/4 around point W. Point K is the midpoint of segment CU. Segments CU and JW intersect at point O. Point N is the midpoint of segment KJ. Find the length of segment NO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RW = 12. Point K is obtained by rotating point R by angle pi/3 about point W. Let O be the reflection of point R across point W. Point V is the reflection of point O across point R. Let H be the midpoint of segment KW. Point L is the reflection of point H across point V. Find the distance between points L 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 AN = 1. Let P be the rotation of point A by angle pi/12 around point N. Let H be the reflection of point P across point N. Let Y be the midpoint of segment NH. Point D is obtained by rotating point N by angle pi/12 about point H. Let I be the midpoint of segment NY. Line Q is the perpendicular bisector of segment DP. Let V be the rotation of point D by angle pi/2 around point P. Let E be the midpoint of segment IV. Line K passes through point V perpendicular to line Q. Point U is the reflection of point D across segment NA. Line L is the perpendicular bisector of segment NA. Point Z is the reflection of point P across line K. Point T is the reflection of point H across line L. Let B be the angle bisector of angle UPZ. Point F is the reflection of point E across line B. What is the length of segment TF? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XB = 1. Point Q is obtained by rotating point B by angle 3pi/4 about point X. Let K be the vector from point X to point B. Let W be point Q translated by vector K. Let R be the rotation of point X by angle pi around point B. Point J is the midpoint of segment XB. Point H is the reflection of point Q across segment XR. Point A is the reflection of point W across point H. What is the length of segment AJ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GK = 3. Point R is obtained by rotating point K by angle pi/4 about point G. Let P be the midpoint of segment GR. Let I be the reflection of point G across point R. Line W passes through points I and R. Point O is the reflection of point P across point G. Let T be the line through points K and P. Let V be the reflection of line T across line W. Line N passes through point O parallel to segment GK. Let Z be the line through point I parallel to line N. Lines Z and V intersect at point X. What is 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}


[User]
YP = 7. Point H is on segment YP. Let X be the perpendicular bisector of segment HY. Point Z is the reflection of point P across line X. Point T is the reflection of point Z across point H. Point U is the midpoint of segment ZH. Point K is the reflection of point U across point H. What is the distance between points T and K? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ZS = 12. Let P be the reflection of point Z across point S. Let G be the reflection of point P across point Z. Let T be the reflection of point G across point P. Let A be the midpoint of segment ZS. Let W be the reflection of point S across point T. Find the distance between points W and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GC = 12. Let W be the rotation of point C by angle 5pi/6 around point G. Point Z is obtained by rotating point C by angle pi/2 about point W. Point H is obtained by rotating point C by angle 11pi/12 about point W. Point X is the reflection of point C across point H. Point F is the midpoint of segment XH. Let N be the line through point H parallel to segment ZG. Point S is the reflection of point F across line N. Point R is the reflection of point H across point S. What is the distance between points R 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 ND = 11. Let Q be the midpoint of segment ND. Point V is obtained by rotating point N by angle pi/12 about point Q. Let K be the midpoint of segment NQ. Let J be the rotation of point D by angle pi/6 around point V. Let O be the line through points J and Q. Let C be the line through point K parallel to segment NV. Lines O and C intersect at point I. Let G be the midpoint of segment NV. What is the length of segment GI? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XL = 8. Point Y is the midpoint of segment XL. Let J be the reflection of point Y across point X. Point B is the midpoint of segment XY. Point Q is the reflection of point B across point L. What is the distance between points Q 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 MR = 1. Let J be the midpoint of segment RM. Let P be the perpendicular bisector of segment MJ. Line A is the perpendicular bisector of segment RJ. Let T be the reflection of line P across line A. Let Q be the midpoint of segment JR. Line F is the perpendicular bisector of segment MQ. Point H is on line F with |HQ| = 12. Line S is the perpendicular bisector of segment HJ. Line K is the reflection of line S across point M. Point C is the reflection of point M across line T. Point L is the reflection of point M across line K. Find the distance between points L and C. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WF = 8. Point P is the reflection of point W across point F. Let Y be the reflection of point W across point P. Let E be the reflection of point Y across point F. Point V is the midpoint of segment EF. Point J is the midpoint of segment YF. Let I be the reflection of point J across point V. Let K be the reflection of point I across point W. Find the distance between points K and J. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KF = 2. Let R be the midpoint of segment KF. Let C be the midpoint of segment RK. Point Z is the midpoint of segment RF. Let P be the rotation of point R by angle 11pi/12 around point C. Let E be the midpoint of segment PZ. Let X be the midpoint of segment ZR. Point M is the midpoint of segment XC. What is the length of segment EM? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AE = 8. Point Q is the midpoint of segment AE. Let F be the reflection of point Q across point A. Let X be the vector from point F to point A. Point B is obtained by rotating point E by angle 5pi/12 about point Q. Let N be the perpendicular bisector of segment BQ. Let C be the rotation of point F by angle pi/12 around point A. Let H be point B translated by vector X. Point R is obtained by rotating point C by angle 3pi/4 about point F. Let P be point E translated by vector X. Point Z is the reflection of point C across line N. Let O be the angle bisector of angle PBZ. Let S be the line through point R parallel to line O. Point U is the reflection of point F across line S. Point W is the midpoint of segment UH. What is the length of segment UW? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let XE = 8. Let Y be the rotation of point E by angle pi/12 around point X. Line V is the angle bisector of angle YEX. Point S is the reflection of point Y across point E. Point G lies on line V at distance 11 from point E. Let K be the reflection of line V through point S. Let N be the reflection of line K across line V. Line R passes through point G perpendicular to line N. Point M is the reflection of point Y across line R. What is the length of segment MS? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GW = 3. Let O be the rotation of point W by angle pi/4 around point G. Let B be the midpoint of segment OG. Let D be the midpoint of segment WG. Line X is the angle bisector of angle OGB. Let A be the midpoint of segment OB. Point R is the reflection of point A across point W. Point V is the midpoint of segment WR. Line L is the angle bisector of angle AGV. Let Q be the reflection of line L across line X. Let U be the reflection of line L across line Q. Let S be the line through points R and W. Point M is the intersection of segments VB and AD. Point J is the intersection of lines S and U. Let P be the reflection of point J across point M. Find the distance between points J 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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UZ = 4. Point K is the reflection of point Z across point U. F is placed 4 units away from Z. Line N is the perpendicular bisector of segment ZK. Let M be the line through point F perpendicular to line N. Let G be the line through point K perpendicular to line M. Point X is the reflection of point U across line G. Point D is the reflection of point X across line N. What is the distance between points D and X? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RF = 3. Line K passes through point F perpendicular to segment FR. Point Q is the midpoint of segment FR. Line O is the angle bisector of angle RQF. Let I be the reflection of line K across line O. Let S be the midpoint of segment QF. Point X is the reflection of point S across line I. Line T is the perpendicular bisector of segment RX. Line V is the reflection of line T across point Q. Point Z is the reflection of point X across line V. What is the distance between points R and Z? Express your answer as a decimal number rounded to 2 decimal places.

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


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Let MC = 9. Point E is the midpoint of segment MC. Let V be the angle bisector of angle CEM. Let U be the angle bisector of angle CME. Point W lies on line V at distance 11 from point C. Point D is obtained by rotating point C by angle 5pi/6 about point W. Line Q passes through points D and C. Point T is the midpoint of segment DW. Point A is obtained by rotating point E by angle pi/3 about point T. Point S lies on line V at distance 8 from point W. Let O be the line through point T parallel to line Q. Line F is the reflection of line U across line O. Let L be the line through point M perpendicular to line F. Point R lies on line L at distance 11 from point A. Point X is the reflection of point R across line O. Point I is the reflection of point W across segment SX. Point P is the reflection of point M across segment SX. What is the distance between points I and P? Express your answer as a decimal number rounded to 2 decimal places.

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


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Let BG = 12. Let A be the midpoint of segment BG. Let W be the rotation of point B by angle 11pi/12 around point A. Point P is the reflection of point W across point G. Let V be the reflection of point G across point P. Point L is the reflection of point V across segment AP. Let Z be the midpoint of segment AP. Point K is the midpoint of segment ZL. What is the length of segment KA? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BZ = 12. Let U be the rotation of point B by angle pi/2 around point Z. Point A is the midpoint of segment ZU. Let V be the rotation of point A by angle pi/2 around point Z. Line H passes through points Z and V. Point C is the reflection of point A across line H. Point R is the midpoint of segment VC. Line X passes through point Z perpendicular to line H. Point E is the reflection of point R across line X. Let M be the midpoint of segment VE. Find the length of segment BM. 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 = 3. Let J be the midpoint of segment SK. Let Z be the line through points S and J. Let W be the rotation of point J by angle 5pi/6 around point K. Point Q is the reflection of point K across segment WJ. Point X is the reflection of point Q across line Z. Point U is the midpoint of segment SQ. Find the length of segment UX. 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 ND = 2. Line F is the perpendicular bisector of segment ND. Let T be the line through point D parallel to line F. Point H is the midpoint of segment ND. Let X be the vector from point H to point N. Point S is the midpoint of segment DH. Line E passes through point N parallel to line T. Point O is the translation of point N by vector X. Point K lies on line E at distance 7 from point N. Point G is the reflection of point S across line F. Point U is the reflection of point K across point D. Let B be the reflection of point U across point G. Let A be the midpoint of segment UO. Find the distance between points B and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ON = 10. Let U be the midpoint of segment ON. Line W is the perpendicular bisector of segment UO. Let B be the rotation of point N by angle 5pi/6 around point U. Let L be the line through point B parallel to line W. Point G is the reflection of point B across segment ON. Let A be the perpendicular bisector of segment GU. Line V is the reflection of line A across point U. Lines V and L intersect at point E. Point K is the reflection of point N across line V. Let Y be the reflection of point E across point G. Point F is the reflection of point K across point B. Point C is the reflection of point B across point F. Find the distance between points C and Y. 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 JY = 2. Let X be the reflection of point Y across point J. Point D is the midpoint of segment JX. Point N is the midpoint of segment YD. Point I is the midpoint of segment XN. Point P is obtained by rotating point J by angle pi about point I. Line Q is the angle bisector of angle XPD. Let O be the rotation of point P by angle 5pi/6 around point D. Point V is the midpoint of segment OI. Let F be the reflection of line Q through point V. Let U be the line through point O parallel to line F. Point A is the intersection of segment XN and line U. Let S be the reflection of point V across point A. What is the distance between points I and S? Express your answer as a decimal number rounded to 2 decimal places.

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


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Let SN = 2. Let E be the rotation of point S by angle 5pi/12 around point N. Let V be the rotation of point N by angle 2pi/3 around point E. Let D be the rotation of point S by angle pi/6 around point E. Let C be the reflection of point N across point V. Let R be the reflection of point E across point N. Point X is the midpoint of segment CR. Point B is the reflection of point D across segment EN. Line W passes through point B perpendicular to segment XR. Point K is the reflection of point V across line W. What is the distance between points K and S? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
DG = 2. Let S be the midpoint of segment DG. Let W be the reflection of point S across point D. Let Y be the angle bisector of angle WDG. Let Z be the reflection of line Y through point S. Line U is the reflection of line Z across line Y. Let B be the perpendicular bisector of segment SD. Line F is the reflection of line B across line Y. Point L is the reflection of point G across line U. Point C is the reflection of point S across line F. Find the distance between points C and L. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XB = 5. Let K be the perpendicular bisector of segment XB. Point Z lies on line K at distance 2 from point B. Point L is the reflection of point Z across segment BX. Let G be the midpoint of segment LB. Let Q be the midpoint of segment GZ. Let F be the vector from point X to point Z. Let T be point Z translated by vector F. Point O is the midpoint of segment QG. Let Y be the midpoint of segment GB. Point P is the midpoint of segment GO. Point S is the midpoint of segment PY. Point J is the reflection of point T across segment BX. Let W be the reflection of point S across point B. Let N be the midpoint of segment JY. Find the length of segment NW. Express your answer as a decimal number rounded to 2 decimal places.

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SE = 12. Point Y is obtained by rotating point E by angle 7pi/12 about point S. Line H passes through points S and E. Line G passes through point E perpendicular to segment YS. Let K be the midpoint of segment YS. Let X be the line through point K parallel to line H. Point Z is the reflection of point S across line G. Let W be the rotation of point Z by angle pi/6 around point Y. Line L passes through point W perpendicular to line X. Point J is the reflection of point S across point E. Point B is the reflection of point J across line L. Point F is the midpoint of segment KJ. Let I be the reflection of point S across point F. What is the length of segment IB? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JZ = 4. Let D be the rotation of point Z by angle pi/2 around point J. Point K is the reflection of point J across point D. Let S be the rotation of point K by angle 3pi/4 around point D. Let L be the rotation of point J by angle pi around point Z. Point E is the reflection of point L across segment SD. Let A be the perpendicular bisector of segment DE. Point B is the reflection of point S across line A. Point G is the midpoint of segment SD. Line C passes through point B perpendicular to segment SD. Point V is the reflection of point K across segment SD. Point X is the reflection of point S across line C. Let U be the angle bisector of angle GDK. Point F is the reflection of point X across line U. Point Q is the reflection of point F across point D. Find the distance between points Q and V. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let YJ = 1. Line N is the perpendicular bisector of segment YJ. Point K is obtained by rotating point Y by angle pi/12 about point J. Let F be the reflection of point K across point J. Let G be the vector from point F to point Y. Point W is the reflection of point F across line N. Point A is the translation of point W by vector G. Point I is the reflection of point A across point F. Point V is the reflection of point I across line N. Let Q be the line through points F and A. Let P be the angle bisector of angle KVA. Point R is the reflection of point J across line P. Point X is the reflection of point R across line Q. Find the distance between points R and X. Express your answer as a decimal number rounded to 2 decimal places.

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


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HM = 5. Point U is obtained by rotating point H by angle pi/3 about point M. Point J is the reflection of point M across point H. Line Q passes through points J and U. Point L is on line Q. Let K be the midpoint of segment JL. What is the measure of angle KJM, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LU = 6. Let J be the midpoint of segment UL. Point Z is the reflection of point J across point L. Let M be the perpendicular bisector of segment UL. Line T is the perpendicular bisector of segment LJ. Point G is the reflection of point L across point Z. Line O passes through point G parallel to line M. Point N is obtained by rotating point Z by angle 5pi/6 about point U. Let S be the midpoint of segment GN. Point A is the reflection of point S across line T. Point W is the reflection of point A across line O. Let C be the reflection of point W across point Z. Find the length of segment GC. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let HO = 8. Line T is the perpendicular bisector of segment HO. Point P is the midpoint of segment OH. Let R be the rotation of point P by angle pi/6 around point O. Line A is the perpendicular bisector of segment HR. Line L is the reflection of line T across line A. Point Q is the intersection of line L and segment HR. Point B is the reflection of point H across line L. Find the distance between points B 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 ZX = 5. Let K be the midpoint of segment XZ. Let L be the reflection of point K across point Z. Line D is the angle bisector of angle ZKX. Point M is on line D. Let C be the reflection of line D through point Z. Point J is the reflection of point X across line C. Let P be the line through points M and L. Point Q is on line P. Point U is the reflection of point Q across point X. Point H is the reflection of point X across segment JU. Point T is the reflection of point Z across segment JU. Find the length of segment TH. 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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WX = 10. Let S be the midpoint of segment XW. Point L is the midpoint of segment SW. Line C is the perpendicular bisector of segment WL. Let N be the rotation of point W by angle 5pi/6 around point S. Line J passes through point X parallel to line C. Point M is the reflection of point N across line J. Point Q is the reflection of point S across line C. Let I be the midpoint of segment MX. What is the distance between points Q and I? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WT = 2. Let E be the vector from point T to point W. Point D is obtained by rotating point W by angle 2pi/3 about point T. Point C is the translation of point W by vector E. Point B is obtained by rotating point C by angle 5pi/12 about point D. Point S is the reflection of point D across point B. Point I is the midpoint of segment WB. Point F is the reflection of point S across point I. Let R be the reflection of point F across point B. Point A is the reflection of point R across segment CW. Find the length of segment SA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AB = 10. Let S be the rotation of point B by angle pi/3 around point A. Point H is the midpoint of segment AS. Let P be the midpoint of segment SH. Let D be the midpoint of segment BS. Let T be the line through point D parallel to segment BP. Point U is the reflection of point P across line T. Point K is the reflection of point U across segment BP. Find the distance between points D and K. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WA = 2. Point U is obtained by rotating point A by angle pi/4 about point W. Let P be the angle bisector of angle WAU. Point G is the reflection of point W across point U. Let N be the angle bisector of angle WGA. Point L is the reflection of point U across point G. Line E passes through point U perpendicular to line P. Line J is the reflection of line E across line N. Segment AG and line J intersect at point B. What is the distance between points B 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 NP = 4. Point K is obtained by rotating point N by angle 11pi/12 about point P. Point Y is the reflection of point K across point N. Let M be the rotation of point Y by angle pi/4 around point K. Let B be the reflection of point P across point Y. Point S is the midpoint of segment YN. Point O is the reflection of point S across segment NM. Find the distance between points O 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 SO = 12. Line W is the perpendicular bisector of segment OS. Point V lies on line W at distance 9 from point O. Line G passes through point S parallel to line W. Point I is the midpoint of segment SO. Point A is the reflection of point O across line G. Point F is the midpoint of segment SA. Point U is the reflection of point I across point V. Find the length of segment FU. 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 S be the perpendicular bisector of segment ZJ. Point G is obtained by rotating point Z by angle 3pi/4 about point J. Let N be the rotation of point G by angle 11pi/12 around point Z. Point V is on line S with |VJ| = 11. Point R is the midpoint of segment VZ. Point X is the midpoint of segment GN. Let W be the reflection of point Z across point X. Point T is the reflection of point R across point X. Find the length of segment TW. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
XZ = 4. Let N be the rotation of point Z by angle pi/12 around point X. Let D be the rotation of point Z by angle 3pi/4 around point N. Point O is the midpoint of segment ZD. Point T is the reflection of point N across segment ZD. Let H be the midpoint of segment DT. Find the length of segment OH. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
IN = 5. Let C be the reflection of point N across point I. Let F be the rotation of point N by angle 11pi/12 around point C. Line O passes through points F and C. Let A be the midpoint of segment FN. Let Z be the angle bisector of angle CAN. Let R be the reflection of point I across point A. Let Q be the reflection of line O across line Z. Point Y is the intersection of lines O and Q. Find the distance between points R 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 ES = 1. Line G is the perpendicular bisector of segment ES. Point Z is obtained by rotating point E by angle 2pi/3 about point S. Let D be the reflection of point Z across point E. Line O passes through point E parallel to line G. Point P is the reflection of point Z across line O. Point T is the reflection of point P across segment EZ. Find the measure of angle DPT, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let KP = 5. Let B be the vector from point P to point K. Let U be the line through points K and P. Let R be the line through point K perpendicular to line U. Point X is obtained by rotating point P by angle 7pi/12 about point K. Line E is the reflection of line R across point P. Let J be the vector from point X to point K. Point Q is the reflection of point P across line E. Point W is the translation of point Q by vector J. Line H passes through points P and X. Point S is the reflection of point P across point K. Let Z be point S translated by vector B. Point A is the reflection of point W across line H. Point N is obtained by rotating point W by angle 5pi/6 about point K. Point F is the reflection of point N across point P. Line M passes through point P parallel to segment ZX. Line M and segment KF intersect at point C. Find the distance between points C and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IX = 7. Let Y be the midpoint of segment XI. Let A be the rotation of point X by angle pi/3 around point I. Point J is the reflection of point X across point A. Point W is the midpoint of segment XA. Point Q is the reflection of point Y across segment IW. What is the measure of angle JWQ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
LG = 4. Let O be the line through points L and G. Let T be the perpendicular bisector of segment GL. Point Z is the reflection of point L across point G. Let V be the reflection of line T through point G. Point E is the intersection of lines O and V. Find the distance between points Z 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 KA = 7. Let C be the midpoint of segment AK. Point Q is the reflection of point C across point A. Let U be the rotation of point K by angle 11pi/12 around point Q. Line H is the angle bisector of angle CAU. Line N is the reflection of line H across point C. Line Z is the perpendicular bisector of segment UC. Let E be the line through point U parallel to line N. Line D is the reflection of line Z across line H. Line M is the reflection of line E across point K. Let B be the line through point Q parallel to line M. Point O is the reflection of point A across line B. Point T is the reflection of point K across point O. Let R be the line through point T parallel to segment AQ. Lines R and D intersect at point P. Find the distance between points O and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OD = 4. Point L is the reflection of point O across point D. Let E be the rotation of point D by angle pi/4 around point L. Point R is obtained by rotating point O by angle 5pi/6 about point D. Let Y be the rotation of point O by angle pi/2 around point D. Let J be the reflection of point O across point E. Line G is the angle bisector of angle RYL. Point X is obtained by rotating point E by angle pi/12 about point J. Line B is the reflection of line G across point R. Line S passes through points L and D. Lines B and S intersect at point Z. Point K is the reflection of point X across point Z. Find the distance between points R and K. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RJ = 11. Let X be the rotation of point J by angle 7pi/12 around point R. Let K be the line through points R and J. Point I is the reflection of point R across point J. Let M be the reflection of point R across point X. Line D passes through point R perpendicular to line K. Point E is the midpoint of segment RX. Point Z is the midpoint of segment EI. Point C is the reflection of point Z across line D. Find the length of segment CM. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let HR = 11. Point M is obtained by rotating point H by angle pi/3 about point R. Point B is the reflection of point R across point M. Point Z is the midpoint of segment MB. Let P be the angle bisector of angle ZMR. Point C is obtained by rotating point Z by angle 7pi/12 about point R. Let D be the midpoint of segment RM. Point V is the reflection of point C across point B. Point U is obtained by rotating point D by angle 5pi/12 about point V. Let L be the reflection of line P through point U. Line X is the reflection of line P across line L. Line N is the reflection of line X across point H. Point S is the reflection of point C across line N. What is the length of segment SD? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
VB = 6. Let H be the midpoint of segment BV. Line L is the perpendicular bisector of segment BH. Point Z lies on line L at distance 5 from point V. Let F be the midpoint of segment HV. Point U is the midpoint of segment FZ. Point E is the reflection of point V across point Z. Point T is the reflection of point U across point E. Point M is the midpoint of segment FE. Find the distance between points T and M. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
DG = 7. Point V is obtained by rotating point D by angle pi/6 about point G. Let Z be the reflection of point G across point V. Let S be the line through point Z perpendicular to segment VZ. Point J is the reflection of point D across segment VZ. Point T is the reflection of point D across line S. Point R is the midpoint of segment VJ. Point N is the midpoint of segment RZ. Find the length of segment NT. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
IP = 4. Let T be the line through points P and I. Let L be the vector from point P to point I. Point V is the reflection of point I across point P. Line W is the perpendicular bisector of segment VP. Let J be point I translated by vector L. Line A is the reflection of line W across point J. Let U be the line through point J parallel to line W. Let M be the reflection of point V across point J. Point S is the intersection of lines T and A. Point K is on line U with |KM| = 1. Line F passes through point S perpendicular to segment IK. Point Q is the reflection of point V across line F. Point C is the reflection of point M across segment IK. What is the distance between points C and Q? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
MN = 6. Let L be the midpoint of segment NM. Let X be the midpoint of segment LN. Let K be the reflection of point X across point L. Point W is the midpoint of segment NK. Find the distance between points W and K. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OC = 1. Point P is the reflection of point C across point O. Point N is the midpoint of segment CO. Let K be the rotation of point P by angle 11pi/12 around point C. Let B be the angle bisector of angle NKC. Point L is the reflection of point P across line B. Let S be the reflection of point L across point K. Let E be the midpoint of segment CK. Point A is the reflection of point E across segment LN. What is the length of segment AS? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FX = 2. Let P be the rotation of point X by angle 7pi/12 around point F. Point V is the midpoint of segment FP. Let E be the rotation of point P by angle 7pi/12 around point V. Point N is the midpoint of segment FX. Let Y be the line through point E perpendicular to segment NE. Point H is the reflection of point P across line Y. Let L be the reflection of point V across point H. What is the length of segment LE? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WU = 11. Point M is the midpoint of segment WU. Let I be the perpendicular bisector of segment WM. Point P is the reflection of point M across point U. Point R is obtained by rotating point U by angle 2pi/3 about point W. Point K is the reflection of point P across line I. Let H be the reflection of line I through point U. Line J is the perpendicular bisector of segment KW. Line F is the reflection of line J across line H. Point Q is the reflection of point R across line F. Point B is the reflection of point K across point Q. Point E is the reflection of point B across point R. Find the length of segment EB. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let CD = 1. Let R be the reflection of point C across point D. Let W be the perpendicular bisector of segment RD. Let K be the reflection of line W through point R. Let G be the midpoint of segment CD. Point N is the midpoint of segment RG. Let O be the reflection of line W across line K. Line F is the reflection of line K across line O. Point J lies on line O at distance 6 from point G. Let H be the line through point N parallel to line F. Let M be the reflection of line H across line O. Point T is the reflection of point J across point G. Point X is the reflection of point T across line M. Find the distance between points N and X. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ZJ = 11. Line T passes through point Z perpendicular to segment JZ. Point O is the midpoint of segment ZJ. Point A is the reflection of point O across line T. Let R be the midpoint of segment ZO. What is the distance between points R and A? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ZO = 7. Let N be the perpendicular bisector of segment OZ. Let M be the line through point O parallel to line N. Point X lies on line M at distance 8 from point Z. Point V is the midpoint of segment XO. Line Q is the reflection of line N across point X. Point W is the reflection of point V across line Q. Point L is the reflection of point V across point X. What is the distance between points L and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SZ = 10. Point U is the midpoint of segment SZ. Let G be the line through point U perpendicular to segment ZU. Point T is the midpoint of segment ZU. Point E is the reflection of point T across line G. Find the distance between points E 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 AZ = 5. Point J is the midpoint of segment ZA. Let D be the reflection of point Z across point A. Let W be the midpoint of segment JD. Let S be the midpoint of segment ZJ. Find the distance between points S and W. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NC = 2. Point I is obtained by rotating point N by angle pi/6 about point C. Point E is the reflection of point C across point I. Let L be the perpendicular bisector of segment EI. Let Z be the midpoint of segment EI. Let B be the perpendicular bisector of segment NE. Let M be the line through point E perpendicular to line L. Let O be the reflection of line B across line M. Point F is obtained by rotating point E by angle pi/6 about point N. Let X be the reflection of point F across point I. Line O and segment FX intersect at point K. Let V be the reflection of point K across point I. What is the distance between points Z 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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CW = 3. Let A be the rotation of point C by angle 5pi/12 around point W. Let R be the rotation of point A by angle pi around point C. Let D be the vector from point W to point R. Let F be the midpoint of segment WA. Let B be point C translated by vector D. Let K be the perpendicular bisector of segment WR. Point P is the reflection of point B across line K. Point T is the reflection of point B across point P. Point X is the reflection of point A across segment BF. Find the distance between points T and X. Express your answer as a decimal number rounded to 2 decimal places.

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


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PH = 5. Let O be the midpoint of segment PH. Point W is obtained by rotating point P by angle pi/3 about point H. Point K is the reflection of point W across point P. Point R is the reflection of point W across point O. Point Z is the reflection of point O across point H. Let L be the angle bisector of angle ROK. Let X be the reflection of line L through point P. Point Q is the reflection of point Z across line X. Find 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}


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Let VT = 5. Let E be the rotation of point V by angle pi around point T. Point I is the midpoint of segment TV. Let H be the line through point I perpendicular to segment EI. Point G is the reflection of point E across line H. Point Q is the midpoint of segment EI. Find the distance between points Q 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 DP = 2. Point F is the reflection of point P across point D. Let X be the reflection of point D across point F. Let U be the midpoint of segment FX. Line J is the perpendicular bisector of segment PU. Point M is the reflection of point X across line J. What is the distance between points M and F? 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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NT = 4. Let J be the midpoint of segment NT. Let Z be the reflection of point J across point T. Let L be the perpendicular bisector of segment NT. Line U is the perpendicular bisector of segment ZT. Let F be the rotation of point N by angle 5pi/12 around point T. Let O be the rotation of point J by angle 5pi/12 around point F. Point M is the reflection of point O across segment NT. Point S is the reflection of point O across line U. Point Y is the midpoint of segment JT. Line G is the reflection of line U across line L. Let C be the line through points Y and S. Lines G and C intersect at point V. What is the length of segment MV? 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 BL = 2. Let X be the reflection of point B across point L. Let Y be the line through points L and X. Line K passes through point L perpendicular to line Y. Point V lies on line K at distance 5 from point L. Let M be the midpoint of segment BV. Point Q is the reflection of point B across point X. Let W be the midpoint of segment MV. Let H be the reflection of point V across point L. Point C is the reflection of point W across point H. Find the distance between points Q 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 OF = 5. Let M be the rotation of point F by angle 2pi/3 around point O. Let R be the midpoint of segment OM. Point Y is the midpoint of segment RO. Point N is the midpoint of segment FR. Point X is the reflection of point Y across segment FR. Find the length of segment NX. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let KV = 4. Let F be the vector from point K to point V. Point R is the translation of point V by vector F. Point M is obtained by rotating point K by angle pi/3 about point V. Line O passes through point M perpendicular to segment KV. Let Q be the reflection of point K across point R. Line P is the perpendicular bisector of segment VQ. Point N is the translation of point Q by vector F. Let E be the midpoint of segment MN. Point A is the midpoint of segment KV. Let T be the line through point K parallel to line P. Line J is the reflection of line O across point E. Point Z is the reflection of point M across line J. Point I is the midpoint of segment ZQ. Point X is the midpoint of segment ZK. Let U be the angle bisector of angle INZ. Point C is the reflection of point I across line U. Point Y is the reflection of point E across line T. Let L be the angle bisector of angle CAY. Point G is the reflection of point E across line L. Point B is the reflection of point G across point I. Find the length of segment BX. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PG = 2. Let E be the vector from point G to point P. Point V is obtained by rotating point G by angle 7pi/12 about point P. Let F be the reflection of point G across point P. Point T is the reflection of point G across point V. Let N be the reflection of point V across point T. Point U is the translation of point T by vector E. Let I be the rotation of point F by angle pi around point U. Line B passes through point I perpendicular to segment UP. Let D be the line through point N parallel to line B. Let R be the line through point T parallel to line D. Point Y is the reflection of point P across line R. Let J be the reflection of point Y across point I. Find the distance between points J and N. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let HI = 11. Line R passes through point I perpendicular to segment IH. Point N is the reflection of point H across line R. Point B is obtained by rotating point H by angle pi about point N. Let V be the rotation of point B by angle 2pi/3 around point I. Let J be the line through point N parallel to line R. Line T is the reflection of line J across point V. Point K is the reflection of point V across segment IH. Point O is the reflection of point K across line J. Let E be the reflection of point N across point O. Point C is the reflection of point E across line T. Let Z be the reflection of point O across point H. 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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RE = 9. Line X passes through points E and R. Let A be the rotation of point E by angle pi/4 around point R. Let Y be the rotation of point A by angle pi/6 around point R. Point L is the midpoint of segment YE. Point C is the reflection of point E across point R. Let P be the midpoint of segment LY. Line G passes through point L parallel to line X. Let I be the reflection of line X across line G. Line S passes through points C and P. Lines S and I intersect at point H. Point K is the reflection of point R across point H. Find the distance between points Y and K. 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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PV = 10. Let C be the line through points V and P. Let F be the line through point V perpendicular to line C. Let K be the reflection of point V across point P. Point R is the midpoint of segment VP. Point X is the reflection of point R across line F. What is the distance between points X and K? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ZX = 5. Let J be the perpendicular bisector of segment XZ. Line S is the reflection of line J across point Z. Point I is the reflection of point X across line S. Line T is the reflection of line S across line J. Let O be the midpoint of segment ZI. Point Y lies on line S at distance 1 from point O. Point A is the reflection of point O across line T. Let U be the line through point A perpendicular to segment XY. Let R be the reflection of line U through point O. Point D is the reflection of point A across line R. Point L is the reflection of point D across segment XY. Let W be the midpoint of segment XD. Point E is the reflection of point L across line J. What is the distance between points E and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VH = 9. Point Q is the reflection of point H across point V. Let B be the rotation of point Q by angle pi/4 around point H. Let S be the rotation of point B by angle 2pi/3 around point V. Let Y be the rotation of point H by angle pi/2 around point S. Point K is obtained by rotating point V by angle 5pi/12 about point H. Point M is the midpoint of segment KY. Point X is the reflection of point M across segment BK. Let E be the midpoint of segment SH. What is the distance between points E and X? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
MV = 3. Let I be the rotation of point M by angle 3pi/4 around point V. Let K be the reflection of point I across point M. Point N is the midpoint of segment KM. Point Y is the reflection of point N across point K. Let X be the perpendicular bisector of segment NM. Line R is the reflection of line X across point Y. Point W is the reflection of point Y across line R. Find the distance between points M and W. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AF = 1. Let Y be the rotation of point F by angle 3pi/4 around point A. Point L is obtained by rotating point A by angle pi/12 about point Y. Line I is the perpendicular bisector of segment LY. Line J is the angle bisector of angle YFL. Line U is the reflection of line J across line I. Line X passes through points A and F. Let M be the perpendicular bisector of segment AL. Point Z is the reflection of point F across line U. Let G be the reflection of point Y across point A. Point D is the reflection of point A across line J. Point V is the reflection of point G across line M. Point C is the reflection of point V across point D. Point H is the reflection of point L across segment YZ. Point N is the reflection of point C across line X. What is the length of segment NH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RX = 5. Let Y be the rotation of point X by angle pi/2 around point R. Point F is obtained by rotating point R by angle pi/3 about point X. Point S is the reflection of point Y across point R. Let T be the vector from point R to point F. Line Q passes through points Y and X. Let G be the reflection of point R across point S. Let A be the reflection of line Q through point S. Point O is the reflection of point F across line A. Let U be point O translated by vector T. Line H passes through points X and G. Let C be the rotation of point S by angle pi/4 around point F. Point W is the reflection of point C across line H. Point D is the reflection of point W across segment YF. Point K is the reflection of point R across segment UD. Find the distance between points O and K. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BK = 9. Let Z be the perpendicular bisector of segment KB. Let R be the midpoint of segment KB. Let E be the reflection of point R across point B. Line Y passes through point K parallel to line Z. Point M is obtained by rotating point B by angle 5pi/12 about point K. Point V is the reflection of point B across line Y. Point I is the reflection of point E across point M. Point W is the reflection of point B across segment IM. Point D is the reflection of point V across segment IM. Let L be the midpoint of segment WD. What is the length of segment IL? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BQ = 3. Let I be the midpoint of segment QB. Point G is obtained by rotating point B by angle 5pi/12 about point I. Let Z be the midpoint of segment BI. Let D be the angle bisector of angle IBZ. Point J is obtained by rotating point Q by angle 5pi/12 about point I. Line O is the perpendicular bisector of segment JB. Line N is the reflection of line D across line O. Let V be the reflection of point G across point B. Point U is the reflection of point V across line N. 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 YK = 3. Line L is the perpendicular bisector of segment YK. Point M is obtained by rotating point Y by angle pi/2 about point K. Let W be the rotation of point M by angle pi/12 around point Y. Point S is the reflection of point K across point W. Point H is the reflection of point W across line L. Point X is the midpoint of segment SM. Point J is the midpoint of segment XH. What is the distance between points J and M? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GW = 6. Let N be the vector from point W to point G. Point J is obtained by rotating point W by angle 3pi/4 about point G. Let P be point J translated by vector N. Let L be the midpoint of segment WJ. Line O is the angle bisector of angle JGW. Let R be the reflection of line O through point P. Point Y is the midpoint of segment GP. Line C passes through point P perpendicular to line R. Point T is the reflection of point L across line C. What is the distance between points Y and T? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PO = 6. Let K be the reflection of point O across point P. Point Z is obtained by rotating point P by angle 11pi/12 about point K. Let T be the line through points Z and P. Let B be the reflection of line T through point K. Let Y be the reflection of line B across line T. Let C be the reflection of line Y across line B. Point J is the midpoint of segment OZ. Point L is the reflection of point Z across line C. Point S is the reflection of point L across segment OZ. What is the length of segment SJ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let PU = 9. Line R is the perpendicular bisector of segment PU. Let E be the reflection of line R through point U. Let J be the rotation of point U by angle pi around point P. Point G is the reflection of point J across line E. Let N be the reflection of point G across point U. Point Y is the reflection of point J across point G. Let X be the midpoint of segment UN. Find the distance between points Y and X. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FP = 10. Let Z be the rotation of point P by angle pi/3 around point F. Point B is the reflection of point P across point Z. Point S is the midpoint of segment FB. Point W is the midpoint of segment PZ. Let L be the midpoint of segment SP. Let Y be the reflection of point W across point F. Point C is the midpoint of segment BZ. Let V be the reflection of point L across point Z. Point K is the midpoint of segment YV. Point J is the reflection of point C across segment YV. Point E is the midpoint of segment SK. Point Q is the midpoint of segment FJ. Line O passes through point Q perpendicular to segment YV. Point D is the reflection of point F across line O. Find the length of segment ED. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let NR = 9. Let T be the rotation of point N by angle 2pi/3 around point R. Point V is the reflection of point T across segment NR. Line O passes through points V and T. Point Y is the reflection of point N across line O. Line U passes through point Y parallel to line O. Let C be the reflection of line U through point N. Point Z is the reflection of point T across line C. Find the distance between points V and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YX = 12. Let D be the reflection of point X across point Y. Let J be the rotation of point D by angle pi around point X. Let F be the reflection of point Y across point J. Point A is obtained by rotating point F by angle 5pi/6 about point Y. Let O be the rotation of point A by angle pi around point J. Line G passes through points D and O. Point H is the reflection of point Y across line G. Point M is obtained by rotating point H by angle 7pi/12 about point J. Let Z be the angle bisector of angle JAH. Let N be the reflection of line Z through point D. Point E is the reflection of point M across line N. Point L is the reflection of point D across point E. Find the distance between points F and L. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HV = 11. Let R be the rotation of point V by angle pi/2 around point H. Let Z be the rotation of point R by angle pi around point V. Let A be the reflection of point H across point V. Let S be the perpendicular bisector of segment VH. Let C be the reflection of point V across point Z. Point E is the reflection of point R across line S. Line L is the reflection of line S across point E. Point O is the reflection of point A across segment CE. Line P is the reflection of line L across point Z. Point D is the reflection of point C across line P. Point G is the reflection of point O across segment RD. What is the length of segment RG? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ST = 7. Point F is the midpoint of segment ST. Line R is the perpendicular bisector of segment TF. Point B is the reflection of point S across line R. Let I be the rotation of point B by angle 3pi/4 around point F. Line U is the reflection of line R across point S. Let O be the midpoint of segment BT. Let W be the line through points F and I. Point H is the intersection of lines U and W. What is the length of segment OH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let MU = 10. Let V be the rotation of point M by angle 2pi/3 around point U. Point S is obtained by rotating point U by angle pi about point V. Let R be the midpoint of segment VU. Let Z be the rotation of point R by angle 11pi/12 around point S. Line H is the perpendicular bisector of segment RS. Line Q passes through point U perpendicular to line H. Point D is the midpoint of segment MS. Point F is the reflection of point D across line Q. Find the distance between points F and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GY = 2. Point Z is the midpoint of segment GY. Point S is obtained by rotating point G by angle pi about point Y. Let D be the perpendicular bisector of segment SY. Line R is the reflection of line D across point Y. Let O be the rotation of point Z by angle 2pi/3 around point S. Let X be the vector from point G to point Y. Let E be the reflection of line R through point O. Let L be the reflection of line E through point S. Let V be point O translated by vector X. Point C is the reflection of point V across line L. Point A is the reflection of point V across line D. Line Q is the angle bisector of angle SCA. Line B is the reflection of line Q across point Z. Let I be the midpoint of segment OZ. Point H is the reflection of point I across line B. What is the distance between points S and H? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IC = 3. Let L be the reflection of point C across point I. Let U be the rotation of point L by angle 2pi/3 around point I. Line K is the perpendicular bisector of segment LU. Let D be the midpoint of segment UC. Point E is the reflection of point D across segment IC. Point O is the reflection of point D across line K. What is the distance between points E and O? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LO = 9. Let H be the midpoint of segment OL. Point J is the midpoint of segment LH. Let W be the reflection of point O across point L. Point U is the reflection of point O across point W. Point C is the midpoint of segment OJ. Point R is the reflection of point J across point L. Let V be the perpendicular bisector of segment RC. Point Q is the reflection of point U across line V. Let N be the reflection of point W across point Q. What is the distance between points R and N? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QN = 3. Line U passes through points N and Q. Point E is obtained by rotating point N by angle pi/6 about point Q. Point J is the reflection of point E across line U. Point P is obtained by rotating point J by angle 11pi/12 about point N. Line V is the perpendicular bisector of segment JP. Point S is the reflection of point E across point Q. Point K is the reflection of point Q across line V. Point R is the reflection of point K across segment PN. Point I is the reflection of point R across segment SP. Find the distance between points I and J. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IC = 3. Point K is the midpoint of segment IC. Let S be the midpoint of segment CK. Point N is obtained by rotating point K by angle pi/6 about point C. Let L be the reflection of point C across point N. Let V be the rotation of point S by angle 3pi/4 around point L. Let M be the reflection of point C across point I. Let Q be the line through point V parallel to segment NM. Point X is the reflection of point S across line Q. Point F is the midpoint of segment NM. Find the distance between points X and F. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RK = 2. Line F is the perpendicular bisector of segment KR. Point E is obtained by rotating point R by angle pi/3 about point K. Line Z passes through points E and K. Point Y lies on line Z. Point J is the midpoint of segment YK. Let W be the perpendicular bisector of segment YK. Line U is the reflection of line Z across point R. Let B be the reflection of line W across line F. Lines B and U intersect at point V. Let X be the reflection of point V across point J. What is the length of segment RX? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VW = 1. Point A is obtained by rotating point W by angle 11pi/12 about point V. Line E is the perpendicular bisector of segment AV. Point Y is obtained by rotating point V by angle 5pi/6 about point A. Line G is the angle bisector of angle YVW. Line L is the reflection of line G across line E. Let Q be the midpoint of segment YW. Point R is the reflection of point Y across line L. What is the distance between points Q 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 HM = 11. Let E be the perpendicular bisector of segment HM. Point Z lies on line E at distance 5 from point M. Let P be the reflection of point Z across point H. Let T be the midpoint of segment ZM. Line J passes through points M and Z. Point Q is the reflection of point P across point T. Point D is the reflection of point Q across line J. Find the distance between points Q and D. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VG = 5. Line N is the perpendicular bisector of segment VG. Point Z is the reflection of point G across point V. Let I be the rotation of point V by angle pi/2 around point G. Let O be the line through point I perpendicular to line N. Line T is the reflection of line N across point I. Line D is the reflection of line T across point Z. Point M is the reflection of point G across line O. Point R is the midpoint of segment ZI. Point X is the reflection of point R across line D. Point K is the reflection of point M across segment ZI. 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]
Let MB = 8. Let A be the rotation of point B by angle pi/12 around point M. Point T is obtained by rotating point A by angle 3pi/4 about point M. Let E be the rotation of point A by angle pi/12 around point M. Let O be the perpendicular bisector of segment TM. Point F is the reflection of point E across line O. Point Z is the reflection of point T across point E. Let I be the reflection of point F across point M. Find the length of segment IZ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HZ = 9. Point J is obtained by rotating point Z by angle pi/4 about point H. Let E be the rotation of point Z by angle 11pi/12 around point J. Let T be the line through points H and E. Point W is the reflection of point J across line T. Let A be the line through point W parallel to line T. Let I be the reflection of line T across line A. Point D is the reflection of point E across line A. Point Y is the reflection of point H across line I. What is the distance between points Y and D? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ZT = 2. Point B is the reflection of point Z across point T. Let W be the reflection of point Z across point B. Line A is the perpendicular bisector of segment ZB. Point V is the reflection of point W across line A. Point Y is on line A with |YV| = 9. Line H is the perpendicular bisector of segment TY. Point M is on line H. Point U is the reflection of point M across line A. Point Q is the reflection of point U across point V. Find the distance between points M and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XZ = 4. Point O is obtained by rotating point X by angle 5pi/6 about point Z. Let V be the midpoint of segment XZ. Point U is obtained by rotating point Z by angle 3pi/4 about point O. Let B be the angle bisector of angle XUZ. Point T is the reflection of point X across point Z. Let E be the reflection of point V across point T. Point M is the reflection of point O across line B. What is the length of segment EM? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let CR = 4. Line M is the perpendicular bisector of segment RC. Point I is obtained by rotating point C by angle 3pi/4 about point R. Point P is the midpoint of segment RI. Line B passes through point I parallel to line M. Point E is obtained by rotating point I by angle 7pi/12 about point P. Let X be the midpoint of segment PC. Point J is the reflection of point X across line B. Point D is the reflection of point E across segment PC. Let N be the midpoint of segment JP. Let Q be the midpoint of segment DE. Find the distance between points N 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 XK = 4. Let R be the line through points X and K. Let G be the line through point X perpendicular to line R. Point J is the reflection of point K across line G. Let B be the reflection of line G through point K. Point N is the reflection of point J across line B. Let F be the midpoint of segment NX. What is the distance between points F 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 ZF = 9. Point X is the midpoint of segment ZF. Point Q is the midpoint of segment FX. Point U is the midpoint of segment QX. Let V be the rotation of point F by angle 5pi/12 around point Z. Point E is obtained by rotating point V by angle pi/4 about point U. Point Y is the midpoint of segment XE. Point A is the midpoint of segment YQ. Let N be the midpoint of segment AV. What is the distance between points E 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 KL = 6. Point M is obtained by rotating point K by angle pi/4 about point L. Point W is the midpoint of segment LK. Point D is the midpoint of segment MK. Point V is the midpoint of segment WD. Point O is the reflection of point V across segment LK. Let S be the reflection of point L across point O. Find the length of segment SV. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RX = 4. Let L be the reflection of point R across point X. Line K is the angle bisector of angle RXL. Point C is obtained by rotating point L by angle pi/12 about point X. Line I passes through point C parallel to line K. Point O is obtained by rotating point L by angle 7pi/12 about point R. Let A be the reflection of point L across point R. Let W be the line through point O parallel to line I. Point U is obtained by rotating point C by angle 7pi/12 about point X. Point S is obtained by rotating point A by angle pi/6 about point U. Point F is the reflection of point S across line K. Let M be the reflection of point F across point C. Line Q passes through point M perpendicular to line W. Point B is the reflection of point U across line Q. What is the length of segment CB? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let FX = 11. Let G be the perpendicular bisector of segment XF. Point V is the reflection of point F across point X. Point O lies on line G at distance 11 from point X. Line B is the perpendicular bisector of segment XV. Point Z is the reflection of point X across point O. Point Q is the reflection of point Z across segment XV. Point E is the reflection of point Z across line G. Point C is the reflection of point Q across line B. Point N is the reflection of point C across segment XV. Point Y is the reflection of point F across point N. Find the length of segment YE. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RW = 9. Let Q be the reflection of point W across point R. Point B is the midpoint of segment WR. Point I is the midpoint of segment BW. Point K is the midpoint of segment RB. Let L be the midpoint of segment KQ. What is the distance between points L and I? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let XV = 12. Point K is obtained by rotating point X by angle pi about point V. Let L be the perpendicular bisector of segment KV. Point C lies on line L at distance 7 from point K. Line F passes through point C perpendicular to line L. Point O is the reflection of point C across segment XV. Let Y be the reflection of point O across point C. Point D is the reflection of point V across line F. Find the length of segment YD. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
WV = 1. Let N be the midpoint of segment WV. Line T is the angle bisector of angle WNV. Point B is on line T with |BN| = 3. Point C is the reflection of point B across segment WV. Let X be the reflection of point B across point C. Point J is the midpoint of segment WX. Point F is the reflection of point J across point V. Find the length of segment FJ. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
BA = 4. Let W be the line through point B perpendicular to segment BA. Point L is obtained by rotating point A by angle 5pi/6 about point B. Line T passes through point L perpendicular to segment BA. Let P be the reflection of line W through point A. Line Q is the angle bisector of angle BLA. Line M is the reflection of line T across line Q. Point V is the reflection of point B across line M. Point E is the reflection of point A across line P. What is the measure of angle VAE, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ZL = 10. Point J is obtained by rotating point L by angle 2pi/3 about point Z. Point R is the reflection of point Z across point J. Let Q be the perpendicular bisector of segment ZR. Point O is the reflection of point L across point R. Let W be the reflection of line Q through point O. Point I is the reflection of point O across line W. Point H is the reflection of point L across line Q. Point M is the reflection of point H across segment JZ. Point G is the reflection of point I across point L. Find the distance between points G and M. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
AN = 7. Let P be the midpoint of segment NA. Point F is the midpoint of segment NP. Point K is the midpoint of segment FP. Let V be the line through point A perpendicular to segment PN. Let X be the midpoint of segment KP. Line E is the angle bisector of angle APX. Point R is the reflection of point F across line E. Point O is the reflection of point F across line V. What is the distance between points O and R? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
GN = 9. Line M is the perpendicular bisector of segment GN. Line V passes through points N and G. Point C lies on line M at distance 9 from point N. Let D be the rotation of point G by angle 7pi/12 around point C. Point T is the reflection of point D across line V. Point S is the reflection of point T across line M. Point P is the reflection of point C across segment DS. What is the length of segment DP? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
QN = 9. Point Y is the midpoint of segment NQ. Let Z be the rotation of point Q by angle pi/3 around point Y. Line M passes through point Q parallel to segment ZY. Let W be the reflection of point N across point Z. Point D is the reflection of point Y across point W. Let O be the rotation of point W by angle pi around point Y. Point I is the midpoint of segment OZ. Line J passes through point D parallel to line M. Point F is the reflection of point O across line J. Point X is the midpoint of segment DI. What is 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}