[User]
MF = 11. Let Z be the midpoint of segment FM. Point E is the reflection of point Z across point F. Let I be the rotation of point F by angle 11pi/12 around point Z. Line P is the perpendicular bisector of segment IE. Point G is the midpoint of segment MI. Let B be the midpoint of segment IZ. Let S be the rotation of point G by angle 3pi/4 around point B. Point O is the reflection of point S across line P. Point V is the midpoint of segment GM. Line U passes through point O parallel to line P. Let K be the perpendicular bisector of segment FV. Let N be the reflection of point V across point M. Let Y be the reflection of line K across line P. Lines U and Y intersect at point A. Point W is the midpoint of segment FN. 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]
HK = 6. Point L is the midpoint of segment KH. Let U be the perpendicular bisector of segment LH. Let Z be the midpoint of segment LH. Point M is the reflection of point Z across point H. Point X is the midpoint of segment LZ. Point W is the reflection of point X across line U. 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]
Let QN = 12. Point H is the midpoint of segment QN. Let T be the rotation of point H by angle pi/2 around point Q. Let Y be the angle bisector of angle TNQ. Point D is on line Y with |DN| = 9. Point E is the reflection of point N across segment TQ. Let L be the reflection of point D across point H. Let V be the reflection of point N across point L. What is the distance between points V and E? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BV = 3. Point S is the reflection of point V across point B. Let F be the rotation of point B by angle pi/2 around point S. Line R passes through point F perpendicular to segment FV. Point C is the reflection of point V across point S. Point H is the reflection of point C across line R. Let P be the midpoint of segment FV. Let J be the midpoint of segment BH. Find the distance between points P 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 YI = 1. Let M be the midpoint of segment IY. Let J be the rotation of point M by angle 7pi/12 around point I. Point Z is the midpoint of segment JM. Let G be the reflection of point I across point Z. Point U is the midpoint of segment ZG. Point D is the midpoint of segment GM. Let B be the reflection of point U across point Z. What is the distance between points B and D? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HX = 11. Line N is the perpendicular bisector of segment XH. Line Y is the reflection of line N across point X. Point W is the midpoint of segment XH. Let T be the midpoint of segment XW. Point O is the reflection of point H across point T. Point J is the midpoint of segment XO. Point E is the reflection of point W across line Y. Find the distance between points E and J. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let XC = 11. Let K be the line through points X and C. Point Y is on line K with |YX| = 1. Line E is the perpendicular bisector of segment CX. Point U lies on line E at distance 2 from point C. Let R be the midpoint of segment XY. Let S be the reflection of point U across point R. Point A is the midpoint of segment YU. Let H be the midpoint of segment SC. What is the length of segment AH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let AO = 7. Let Y be the reflection of point O across point A. Let G be the midpoint of segment AY. Point T is the reflection of point G across point O. Point Z is the midpoint of segment TY. Find the distance between points Y and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let MS = 6. Let R be the reflection of point S across point M. Let Q be the perpendicular bisector of segment RS. Point C is the reflection of point M across point S. Let B be the line through point C parallel to line Q. Let X be the reflection of point S across point C. Point L is obtained by rotating point S by angle 2pi/3 about point R. Let P be the reflection of point R across point X. Point Z is the reflection of point R across line B. Point H is the reflection of point P across segment LC. Find the distance between points H and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
MW = 11. Let Z be the midpoint of segment MW. Let R be the line through point M perpendicular to segment MW. Point N is on line R with |NZ| = 9. Point H is the reflection of point N across segment MW. Line A passes through point W perpendicular to segment HW. Point L is the reflection of point N across line A. Let U be the midpoint of segment HW. What is the length of segment LU? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BC = 5. Let E be the vector from point C to point B. Let L be the midpoint of segment CB. Point F is the translation of point L by vector E. Let Y be the perpendicular bisector of segment LF. Point X lies on line Y at distance 10 from point F. Point G is the midpoint of segment CL. Line N is the angle bisector of angle XLC. Let Q be the midpoint of segment XC. Point U is the midpoint of segment GQ. Let A be the midpoint of segment LU. Point S is the midpoint of segment GF. Point R is the reflection of point S across line N. Point P is the reflection of point R across segment AG. Find the distance between points A and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let IG = 7. Let Z be the rotation of point I by angle 5pi/6 around point G. Line C passes through points Z and G. Line Y passes through points Z and I. Point A lies on segment IG. Point F is the reflection of point A across line Y. Point X is the reflection of point F across line C. Find the distance between points F and X. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IO = 5. Let U be the rotation of point O by angle 2pi/3 around point I. Let K be the midpoint of segment UI. Line A is the perpendicular bisector of segment OI. Point H is the reflection of point K across line A. Find the distance between points H and U. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JD = 7. Let C be the perpendicular bisector of segment DJ. Let V be the reflection of line C through point D. Point N is the reflection of point J across line V. Let Y be the midpoint of segment JD. What is the distance between points N and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PY = 2. Point N is obtained by rotating point P by angle pi/2 about point Y. Point Q is obtained by rotating point Y by angle pi/12 about point P. Point E is the midpoint of segment QY. Line H passes through points N and Q. Line U passes through point E perpendicular to line H. Point R is the reflection of point Q across line U. What is the length of segment RP? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AT = 1. Point Z is obtained by rotating point A by angle 7pi/12 about point T. Point G is obtained by rotating point Z by angle 11pi/12 about point A. Point C is obtained by rotating point A by angle pi/6 about point T. Let U be the rotation of point G by angle pi/3 around point C. Let K be the rotation of point Z by angle 3pi/4 around point A. Point R is obtained by rotating point C by angle 7pi/12 about point A. Let J be the rotation of point T by angle pi/2 around point K. Let L be the rotation of point J by angle 5pi/6 around point A. Line W is the angle bisector of angle GLR. Let F be the line through point L perpendicular to line W. Let X be the reflection of point C across point U. Let S be the line through point K perpendicular to line F. Point O is the reflection of point X across line S. Point D is the reflection of point U across point L. Point N is the midpoint of segment DO. What is the distance between points N 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 CR = 8. Point D is the reflection of point C across point R. Let Y be the rotation of point D by angle 3pi/4 around point C. Let H be the perpendicular bisector of segment CR. Line G is the reflection of line H across point R. Point W is the midpoint of segment RY. Point B is the midpoint of segment YD. Let N be the reflection of point W across point Y. Point T is the midpoint of segment CN. Point X is the reflection of point T across line G. Point J is the reflection of point X across segment CN. Find the length of segment BJ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RN = 5. Point D is obtained by rotating point N by angle 5pi/6 about point R. Let H be the rotation of point D by angle 3pi/4 around point N. Let Z be the line through points D and H. Point P is the midpoint of segment DH. Let A be the reflection of line Z through point N. Line C passes through points P and R. Let Y be the reflection of point H across point R. Point X is the intersection of lines A and C. Find the length of segment YX. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BL = 6. Point C is the midpoint of segment LB. Let J be the rotation of point B by angle 5pi/6 around point C. Point Q is obtained by rotating point C by angle 2pi/3 about point J. Point M is the reflection of point J across point L. Point S is the reflection of point Q across segment MB. Let W be the reflection of point C across point M. Find the distance between points W 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 JZ = 7. Let Y be the reflection of point J across point Z. Let V be the midpoint of segment ZY. Let Q be the reflection of point Y across point J. Point N is the midpoint of segment VJ. Point A is the reflection of point N across point Q. What is the distance between points A 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 QJ = 4. Point G is the midpoint of segment JQ. Point S is the midpoint of segment GQ. Let N be the rotation of point J by angle 3pi/4 around point G. Point U is the midpoint of segment NS. Line C passes through points G and N. Point R is the reflection of point U across line C. Point X is the reflection of point R across segment QN. Find the distance between points X and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XS = 1. Let J be the midpoint of segment SX. Point N is obtained by rotating point X by angle 5pi/6 about point J. Let C be the midpoint of segment JX. Let W be the midpoint of segment CN. Point G is the midpoint of segment SJ. Let K be the perpendicular bisector of segment JW. Let L be the midpoint of segment SN. Let R be the reflection of line K through point G. Line Q passes through point J parallel to line R. Point I is the reflection of point G across line Q. Find the distance between points L 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 WV = 5. Let J be the perpendicular bisector of segment WV. Point U is the midpoint of segment VW. Let S be the reflection of point U across point V. Point A is the reflection of point S across line J. Point T is the midpoint of segment VS. Let O be the reflection of point A across point T. What is the distance between points U and O? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VN = 7. Point H is the midpoint of segment VN. Let B be the vector from point V to point H. Let P be the rotation of point N by angle pi/2 around point H. Let A be the midpoint of segment HV. Point Z is the translation of point N by vector B. Point R is the reflection of point A across point Z. Line Q passes through points P and H. Line I passes through point N perpendicular to line Q. Point W is the reflection of point N across segment VP. Let S be the line through point R perpendicular to line I. Line E passes through point H parallel to segment VP. Let X be the reflection of line E through point W. Line U is the reflection of line S across line Q. Point J is the midpoint of segment WA. Let Y be the line through point R parallel to line X. Lines Y and U intersect at point O. What is the distance between points J and O? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
UI = 2. Point E is obtained by rotating point I by angle pi/12 about point U. Let S be the vector from point U to point E. Point K is the reflection of point I across point E. Let J be the midpoint of segment IE. Let X be the reflection of point U across point E. Point Q is obtained by rotating point E by angle 5pi/6 about point U. Let Y be point Q translated by vector S. Let P be point J translated by vector S. Point W is the reflection of point Y across point E. Let V be the line through points Q and W. Point L is the reflection of point X across line V. Point N is the midpoint of segment KP. What is the length of segment LN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AZ = 6. Point K is the reflection of point Z across point A. Let I be the line through points A and Z. Let C be the rotation of point K by angle pi/12 around point A. Line W passes through point C perpendicular to line I. Point Y is obtained by rotating point Z by angle pi about point C. Point L is the reflection of point Z across line W. Point U is obtained by rotating point Z by angle 5pi/12 about point L. Let V be the reflection of point Y across point K. Let B be the midpoint of segment VC. Let O be the midpoint of segment AU. Let S be the reflection of point B across point O. Find the distance between points B and S. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YA = 8. Let Q be the rotation of point A by angle 7pi/12 around point Y. Let G be the reflection of point Y across point A. Line H passes through points G and Y. Let F be the line through point Q parallel to line H. Point T is the reflection of point A across line F. Let P be the midpoint of segment QG. Find the distance between points T and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let UQ = 4. Let Y be the reflection of point Q across point U. Line J is the angle bisector of angle QUY. Point O is obtained by rotating point Y by angle 7pi/12 about point U. Let X be the reflection of point O across point Y. Let B be the reflection of line J through point X. Line V is the perpendicular bisector of segment QO. Let L be the reflection of point X across point O. Line N passes through point Q parallel to line V. Let A be the vector from point O to point X. Lines B and N intersect at point C. Point E is the translation of point C by vector A. Point R is the reflection of point E across line B. Point T is the reflection of point E across point R. Line F passes through point T perpendicular to segment YL. Point H is the reflection of point C across line F. Find the distance between points U and H. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IU = 1. Line Z passes through points U and I. Line Q passes through point U perpendicular to line Z. Point L is the reflection of point U across point I. Point N is on line Q with |NU| = 5. Point G is the midpoint of segment LI. Line R passes through points N and L. Point S is the reflection of point G across line Q. Point T is the reflection of point S across line R. Point W is the midpoint of segment TI. 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]
Let PS = 11. Let C be the reflection of point P across point S. Point U is obtained by rotating point C by angle 3pi/4 about point S. Point E is obtained by rotating point C by angle 5pi/12 about point P. Point A is the reflection of point P across point E. Line R is the perpendicular bisector of segment CS. Point X is the midpoint of segment CS. Let I be the midpoint of segment CX. Let Y be the perpendicular bisector of segment SU. Point O is obtained by rotating point A by angle pi/3 about point P. Point V is obtained by rotating point I by angle pi about point E. Let J be the reflection of point A across point P. Line G is the reflection of line Y across line R. Point L is the reflection of point V across segment OJ. Point N is the reflection of point L across line G. What is the distance between points N and E? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OP = 3. Point K is the midpoint of segment OP. Let T be the midpoint of segment OK. Point V is the reflection of point T across point O. Line W is the perpendicular bisector of segment VK. Let A be the reflection of line W through point K. Point Q is the reflection of point O across line A. What is the distance between points Q and T? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OH = 9. Let A be the midpoint of segment HO. Line S is the perpendicular bisector of segment OA. P is constructed so that HP = 9. Let R be the line through point P parallel to line S. Let J be the rotation of point A by angle pi/6 around point H. Let N be the line through point O parallel to line R. Point T is the reflection of point J across segment HO. Point M is the midpoint of segment HJ. Let F be the line through point T parallel to line N. Point Q is the reflection of point M across segment HO. Point X is the midpoint of segment QJ. Segment HO and line F intersect at point Z. What is the measure of angle XZT, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let JL = 4. Line O is the perpendicular bisector of segment JL. Let S be the vector from point J to point L. Let M be the midpoint of segment LJ. Point N is the translation of point L by vector S. Point P is the reflection of point N across line O. Point R is obtained by rotating point L by angle 11pi/12 about point N. Let X be the line through point P perpendicular to segment RM. Point H is the reflection of point M across line X. Let F be the midpoint of segment RM. Find the distance between points H and F. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BS = 7. Point F is the reflection of point B across point S. Let H be the rotation of point B by angle 3pi/4 around point S. Line M is the perpendicular bisector of segment BF. Let Y be the vector from point S to point B. Let L be the line through point H parallel to segment BF. Point T is obtained by rotating point B by angle 7pi/12 about point S. Let V be the reflection of point H across point F. Let W be the line through points T and V. Let P be point V translated by vector Y. Point D is the intersection of lines L and W. Point K is the reflection of point P across segment BF. Point J is the reflection of point K across point B. Point O is the reflection of point J across line M. Point A is the reflection of point D across point J. Find the length of segment AO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KW = 5. J is constructed so that WJ = 2. Let Z be the line through points J and W. Let R be the reflection of line Z through point K. Point P is the reflection of point J across line R. Point M is the reflection of point W across segment PK. What is the length of segment MK? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let SI = 4. Let E be the rotation of point S by angle pi/6 around point I. Let G be the angle bisector of angle IES. Point C is the reflection of point S across line G. Point J is the reflection of point C across segment SI. What is the length of segment JC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BY = 11. Let X be the midpoint of segment YB. Let E be the midpoint of segment BX. Let R be the midpoint of segment XE. Let J be the reflection of point B across point R. Point Z is obtained by rotating point R by angle 7pi/12 about point E. Line V is the angle bisector of angle XZB. Let L be the line through point B parallel to line V. Let D be the line through point R parallel to line L. Let F be the reflection of line D across line L. Line A is the reflection of line D across line V. Line N passes through points Z and J. Point Q is the intersection of lines F and N. Point W is the reflection of point Q across line A. Find the length of segment QW. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WE = 12. Let G be the reflection of point W across point E. Let B be the vector from point E to point G. Point S is the translation of point G by vector B. Point C is obtained by rotating point S by angle 5pi/12 about point W. Let J be the perpendicular bisector of segment GW. Let I be the reflection of line J through point S. Point M is the reflection of point E across point C. Point Y is the reflection of point S across line I. Find the distance between points Y 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 NO = 10. Let Y be the rotation of point N by angle pi/4 around point O. Let H be the reflection of point Y across point N. Let E be the perpendicular bisector of segment OY. Point F is the reflection of point N across line E. Point J is the midpoint of segment FH. Let B be the midpoint of segment JO. What is the length of segment BY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
UP = 8. Point M is the reflection of point U across point P. Line H passes through point M perpendicular to segment UM. Point T is the reflection of point P across line H. Let W be the midpoint of segment UP. Find the distance between points W and T. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let LZ = 2. Point O is obtained by rotating point Z by angle pi about point L. Point U is obtained by rotating point L by angle pi/2 about point Z. Point K is the midpoint of segment UZ. Line G is the perpendicular bisector of segment OU. Point X lies on segment ZL. Let F be the perpendicular bisector of segment UZ. Line W is the reflection of line G across line F. Line I is the reflection of line F across line G. Lines I and W intersect at point H. Let R be the line through point U parallel to line W. Let P be the reflection of line F through point X. Point N is the intersection of lines P and R. Point M is the reflection of point K across point H. What is the distance between points M and N? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XT = 2. Let A be the vector from point T to point X. Let P be the midpoint of segment XT. Point S is the translation of point X by vector A. Point W is obtained by rotating point P by angle pi/3 about point S. Let G be the line through points T and X. Let H be the line through point S perpendicular to line G. Point E is obtained by rotating point W by angle 5pi/12 about point S. Let R be the perpendicular bisector of segment WP. Point F is obtained by rotating point E by angle pi/3 about point S. Point I is the reflection of point E across segment FT. Let Q be the midpoint of segment FT. Point D is the reflection of point Q across line H. Point Z is the reflection of point D across line R. What is the distance between points I and Z? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let KV = 11. Point N is obtained by rotating point K by angle 5pi/6 about point V. Let G be the perpendicular bisector of segment VN. Line H passes through point N parallel to line G. Let P be the line through point K perpendicular to line H. Let J be the line through point V perpendicular to line P. Point D is the reflection of point N across line J. What is the distance between points D and N? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SD = 2. Let E be the midpoint of segment SD. Let R be the perpendicular bisector of segment SD. Let K be the midpoint of segment SE. Point T is the reflection of point D across point S. Let U be the midpoint of segment KE. Point F is the midpoint of segment TE. Point H is the reflection of point U across line R. Let I be the midpoint of segment EU. Point V is the midpoint of segment ED. Point L is obtained by rotating point V by angle 7pi/12 about point U. Let J be the rotation of point I by angle 11pi/12 around point T. Point Y is obtained by rotating point K by angle pi/12 about point J. Let X be the reflection of point Y across point H. Let A be the midpoint of segment LX. Point Z is the reflection of point F across segment LX. What is the distance between points A and Z? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SD = 4. Point A is obtained by rotating point D by angle 7pi/12 about point S. Point Q is the midpoint of segment AD. Point X is the reflection of point S across point Q. Line Y is the angle bisector of angle XQD. Point M is the reflection of point D across line Y. What is the length of segment MQ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WU = 7. Let G be the perpendicular bisector of segment UW. Point R lies on line G at distance 3 from point U. Point H is the reflection of point R across point U. Point Q is the midpoint of segment UW. Let X be the perpendicular bisector of segment QR. Line P is the reflection of line X across point Q. Line T passes through point W parallel to line P. Line L is the reflection of line T across line X. Point K is the reflection of point U across segment QH. Point E is the reflection of point H across line L. Find the distance between points K and E. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FB = 7. Let E be the rotation of point F by angle 5pi/6 around point B. Let P be the rotation of point B by angle 3pi/4 around point E. Let U be the rotation of point F by angle pi/6 around point E. Let A be the reflection of point F across point P. Point N is the midpoint of segment FU. Let L be the midpoint of segment AN. Let J be the reflection of point L across point F. What is the distance between points J 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 EB = 10. Let Z be the rotation of point B by angle pi/6 around point E. Let O be the reflection of point B across point E. Let H be the reflection of point O across point Z. Point C is the reflection of point E across point H. Let A be the vector from point O to point Z. Let U be the perpendicular bisector of segment ZO. Let Q be point B translated by vector A. Point M is obtained by rotating point Z by angle 3pi/4 about point H. Line D passes through point M parallel to line U. Let R be the midpoint of segment HZ. Line G passes through point C perpendicular to segment HZ. Point F is the reflection of point R across line G. Line D and segment CQ intersect at point K. Find the distance between points F 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 XQ = 6. Point P is obtained by rotating point X by angle 7pi/12 about point Q. Let Z be the perpendicular bisector of segment QX. Line H passes through point P perpendicular to line Z. Point N is the reflection of point P across line Z. Point A is the midpoint of segment QN. Point S is the reflection of point A across line H. Let O be the reflection of point A across point S. Point W is the reflection of point O across line Z. Let D be the midpoint of segment WA. Find 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 ZC = 5. Let J be the midpoint of segment ZC. Line F is the perpendicular bisector of segment ZC. Let N be the rotation of point Z by angle pi/12 around point C. Point M is the reflection of point J across point N. Let D be the line through point C parallel to line F. Let G be the angle bisector of angle ZMN. Point S is the reflection of point N across segment ZC. Let I be the line through point Z parallel to line G. Lines I and D intersect at point H. Let Q be the midpoint of segment SZ. Let E be the midpoint of segment QJ. Find the length of segment HE. 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 GI = 10. Line N passes through points I and G. Let A be the perpendicular bisector of segment IG. Let X be the vector from point G to point I. Point Z is obtained by rotating point I by angle pi/4 about point G. Let P be the reflection of line A through point I. Let Y be point Z translated by vector X. Let H be the reflection of point G across point Y. Let Q be point H translated by vector X. Let F be the rotation of point Q by angle 5pi/12 around point G. Let V be the rotation of point H by angle 2pi/3 around point Z. Let U be the midpoint of segment FV. Let W be the line through point H parallel to line N. Point T is the intersection of lines P and W. Point B is the reflection of point Q across segment UF. What is the distance between points T and B? 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 KP = 12. Let X be the vector from point P to point K. Let I be the rotation of point K by angle 3pi/4 around point P. Line W passes through points K and I. Line M passes through points K and P. Point J is the reflection of point I across point K. Point D is the translation of point K by vector X. Let G be the line through points D and J. Line R is the reflection of line G across line W. Point F is the reflection of point J across line M. Point H is the reflection of point I across line R. Let B be the midpoint of segment FH. Point A is the reflection of point B across line G. What is the distance between points A 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 BF = 11. Line A is the perpendicular bisector of segment BF. Let J be the line through point F parallel to line A. Let O be the reflection of point F across point B. Let V be the rotation of point B by angle 5pi/6 around point O. Let U be the reflection of point B across point O. Point R is obtained by rotating point U by angle pi/6 about point O. Let C be the midpoint of segment RU. Point I is the reflection of point B across line J. Let H be the midpoint of segment BV. Line J and segment IC intersect at point K. What is the length of segment HK? 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 MT = 10. Point L is the reflection of point T across point M. Let N be the perpendicular bisector of segment ML. Let X be the perpendicular bisector of segment MT. Point W is the midpoint of segment MT. Point O is the reflection of point T across line N. Point U is on line X with |UO| = 3. Let J be the midpoint of segment MU. Let V be the midpoint of segment OU. Line D is the perpendicular bisector of segment WM. Point P is the reflection of point V across line D. Let Q be the reflection of point W across point J. What is the distance between points P and Q? 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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BV = 7. Point Z is the midpoint of segment BV. Let N be the rotation of point B by angle 2pi/3 around point Z. Let E be the vector from point B to point V. Line J passes through points V and N. Let I be point V translated by vector E. Line Y passes through point V perpendicular to line J. Point O is on line Y with |ON| = 8. Point D is the reflection of point I across point O. Point X is the midpoint of segment OB. Let C be the reflection of point N across point X. Point Q is the midpoint of segment BD. 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}


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Let AX = 4. Let Q be the rotation of point A by angle pi/6 around point X. Point F is the reflection of point X across point A. Point G is the reflection of point Q across point X. Let H be the reflection of point G across point F. Point D is the midpoint of segment XQ. Point W is the midpoint of segment FQ. Point K is the midpoint of segment WX. Let V be the midpoint of segment KA. Let E be the reflection of point H across point D. Let R be the reflection of point E across point W. Find the distance between points R and V. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VM = 7. Let S be the midpoint of segment VM. Line J passes through points S and M. Let R be the line through point V perpendicular to line J. Point N is the reflection of point M across line R. Find the distance between points N and V. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
TZ = 10. Let Y be the rotation of point T by angle 11pi/12 around point Z. Let F be the midpoint of segment TZ. Line B is the perpendicular bisector of segment TF. Point H is on line B with |HZ| = 10. Point S is the reflection of point Y across line B. Point W is the midpoint of segment YS. Point Q is the reflection of point Y across segment TZ. Let N be the line through point H parallel to segment TZ. Point O is the reflection of point W across line N. Find the length of segment OQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
TH = 7. Let W be the perpendicular bisector of segment HT. Line Q is the reflection of line W across point T. Let U be the reflection of point H across point T. Line A is the reflection of line Q across line W. Point S lies on line A at distance 10 from point T. Point Y is the midpoint of segment SU. Line L passes through points Y and H. Let D be the reflection of line W across line L. Point F is the intersection of line D and segment TS. Point V is the reflection of point S across line L. What is the distance between points V and F? Express your answer as a decimal number rounded to 2 decimal places.

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


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DK = 4. P is constructed so that KP = 2. Point J is obtained by rotating point P by angle 5pi/12 about point K. Point E is the reflection of point P across point K. Point F is the reflection of point J across point P. Point Q is the reflection of point F across segment EK. Find 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}


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Let RK = 9. Let M be the rotation of point K by angle pi/2 around point R. Let O be the rotation of point R by angle pi/4 around point M. Point J is the midpoint of segment KR. Let D be the midpoint of segment KO. Point C is the reflection of point J across point D. What is the length of segment CJ? 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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WA = 3. Let B be the reflection of point A across point W. Let R be the reflection of point A across point B. Let U be the line through point B perpendicular to segment AW. Let D be the reflection of line U through point R. Point P is on line U. Point O is the reflection of point P across line D. What is 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]
MN = 8. Let H be the line through points M and N. Line J passes through point M perpendicular to line H. Let U be the rotation of point M by angle pi/12 around point N. Point G is obtained by rotating point U by angle pi/4 about point M. Let V be the vector from point U to point N. Let R be the line through point G parallel to line J. Point C is the translation of point G by vector V. Let F be the rotation of point M by angle 11pi/12 around point N. Point W is obtained by rotating point F by angle 5pi/12 about point C. Point D is the reflection of point U across point W. Point I is the reflection of point D across line R. Point O is the reflection of point C across point M. Point A is the reflection of point W across point I. What is the length of segment OA? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let MS = 12. Let C be the perpendicular bisector of segment SM. Line J passes through point M parallel to line C. Point Y is obtained by rotating point S by angle pi about point M. Let G be the reflection of line J through point S. Let I be the midpoint of segment MY. Point W is the reflection of point I across line G. Let P be the reflection of point I across point W. Find the distance between points I and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QV = 1. Let G be the line through points Q and V. Point B is on line G with |BV| = 7. Point E is obtained by rotating point V by angle pi/3 about point Q. Line A is the perpendicular bisector of segment VQ. Line X is the perpendicular bisector of segment EQ. Line T passes through point Q parallel to line X. Let Y be the reflection of point B across point E. Point W is the reflection of point Y across line T. Point P is the reflection of point W across line A. Find the measure of angle YWP, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BE = 4. Point G is the reflection of point B across point E. Let A be the rotation of point E by angle 11pi/12 around point B. Let N be the midpoint of segment BA. Let U be the midpoint of segment GN. Point L is the midpoint of segment AU. What is the length of segment NL? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ZU = 1. Point Q is the reflection of point U across point Z. Point S is obtained by rotating point Z by angle pi/12 about point Q. Point L is the reflection of point S across segment UZ. Let I be the midpoint of segment UZ. Let K be the midpoint of segment IL. Let N be the reflection of point K across point S. What is the distance between points Z 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 WI = 9. Point M is obtained by rotating point I by angle 3pi/4 about point W. Let G be the angle bisector of angle WMI. Line D passes through point I parallel to line G. Point E is obtained by rotating point I by angle 7pi/12 about point W. Line Z is the reflection of line D across line G. Let X be the midpoint of segment IM. Let O be the rotation of point E by angle 7pi/12 around point X. Point J is the reflection of point E across line Z. Point Y is the reflection of point J across segment IM. Let V be the midpoint of segment OI. Let L be the midpoint of segment MV. Find the length of segment LY. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let CK = 5. Point O is the reflection of point K across point C. Let Z be the perpendicular bisector of segment OK. Line M passes through point O parallel to line Z. Point R is the reflection of point C across line M. Point V is the reflection of point R across line Z. Point I is the midpoint of segment OC. Let N be the reflection of point V across point I. Point W is the reflection of point N across point V. Point A is the midpoint of segment OW. Point E is the reflection of point V across point A. What is the distance between points C and E? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let AB = 1. Let AD = 4. Point N is obtained by rotating point A by angle 5pi/6 about point D. Line G is the perpendicular bisector of segment DN. Let H be the line through point B parallel to line G. Point P is on line H. Point J is the reflection of point D across point N. Let S be the reflection of line G through point P. Point O lies on line S. Point Z is the reflection of point D across segment OJ. Find the distance between points J and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let NM = 12. Line O passes through points N and M. Let A be the line through point M perpendicular to line O. Let Z be the rotation of point N by angle 7pi/12 around point M. Let F be the reflection of line A through point Z. Let L be the line through point Z parallel to line F. Point B is the reflection of point M across line L. Let Q be the reflection of point B across point Z. Point G is the reflection of point Q across point N. Find the distance between points Z and G. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let PR = 10. Let Q be the reflection of point P across point R. Let Z be the rotation of point P by angle 5pi/6 around point Q. Let U be the midpoint of segment ZQ. Let O be the line through points Q and U. Point C is the reflection of point R across point Q. Point X is on line O. Point F is the reflection of point R across point X. Point G is the midpoint of segment CF. Find the measure of angle UXG, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let IZ = 12. Point R is the reflection of point Z across point I. Point M is the reflection of point R across point Z. Point O is the midpoint of segment ZI. Point C is the midpoint of segment MO. Find the distance between points C 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 NA = 1. Let K be the rotation of point A by angle 11pi/12 around point N. Point V is the midpoint of segment NK. Let U be the midpoint of segment VK. Point P is the reflection of point U across segment KA. What is the length of segment PK? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RL = 7. Line K is the perpendicular bisector of segment RL. Line J passes through point R parallel to line K. Let N be the reflection of point L across point R. Let A be the reflection of line J across line K. Let D be the line through point N parallel to line K. Point G is on line D with |GR| = 11. Let P be the reflection of point L across point G. Point U is the reflection of point P across line A. Let H be the reflection of point N across point U. Let Z be the reflection of point H across point N. Find the distance between points H and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let IH = 3. Let W be the midpoint of segment HI. Let L be the reflection of point W across point H. Point O is obtained by rotating point W by angle 7pi/12 about point I. Let P be the angle bisector of angle OIH. Point Q is the reflection of point H across line P. Point F is the reflection of point Q across segment OW. What is the length of segment FL? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JP = 4. Line Y is the perpendicular bisector of segment PJ. Let E be the midpoint of segment JP. Let H be the midpoint of segment EP. Let M be the midpoint of segment HE. Point B is obtained by rotating point E by angle 2pi/3 about point M. Point G is the reflection of point M across line Y. Point V is the reflection of point B across segment HE. Point S is the reflection of point H across segment GV. Point X is the midpoint of segment GV. What is the distance between points S 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 JA = 9. Let F be the rotation of point J by angle 2pi/3 around point A. Line Z is the angle bisector of angle FAJ. Let Q be the line through point J parallel to line Z. Let C be the midpoint of segment AF. Point O is the reflection of point C across point J. Let P be the reflection of point O across point F. Point G is the reflection of point P across line Q. Find the distance between points G and O. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ZA = 9. Let N be the midpoint of segment AZ. Let E be the reflection of point N across point A. Point B is the midpoint of segment ZN. Point F is obtained by rotating point A by angle pi/6 about point E. Point P is the midpoint of segment BF. Let T be the line through points P and Z. Line S passes through point E parallel to line T. Point U is the reflection of point Z across line S. Find the length of segment UZ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let PD = 10. Point I is the reflection of point D across point P. Let F be the rotation of point P by angle pi/12 around point I. Line N is the angle bisector of angle FDP. Line Z passes through point D perpendicular to line N. Point V is the reflection of point I across line N. Line E passes through points I and P. Let C be the vector from point I to point D. Point U is the translation of point F by vector C. Point A is the reflection of point U across line Z. Let G be the line through point A perpendicular to line N. Point J is obtained by rotating point P by angle 3pi/4 about point F. Point K is the midpoint of segment JU. Let R be the midpoint of segment KJ. Let L be the reflection of point D across point I. Point O is the reflection of point A across segment VR. Let M be the line through point O perpendicular to line E. Line H is the reflection of line M across line G. Line H and segment VR intersect at point T. Find the length of segment TL. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IJ = 4. Let F be the vector from point I to point J. Let Q be point J translated by vector F. Let U be the rotation of point J by angle 3pi/4 around point Q. Let T be the reflection of point Q across point I. Let E be the rotation of point Q by angle pi/6 around point I. Let P be the rotation of point E by angle pi/2 around point J. Let R be the reflection of point U across point Q. Point C is the midpoint of segment PT. Point B is the reflection of point U across segment RC. Point V is the reflection of point B across segment UE. Point W is the reflection of point V across segment RC. 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]
DU = 12. Point H is obtained by rotating point D by angle 5pi/12 about point U. Let A be the midpoint of segment DH. Let I be the midpoint of segment AD. Point E is the midpoint of segment UI. Find the distance between points A 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 UJ = 8. Let H be the reflection of point J across point U. Let V be the rotation of point U by angle 11pi/12 around point J. Point P is obtained by rotating point U by angle 2pi/3 about point H. Line M passes through point J perpendicular to segment UH. Point E is the reflection of point P across line M. Point X is the reflection of point V across segment UH. Line G passes through point X parallel to segment UH. Let Y be the angle bisector of angle EHP. Point Z is the reflection of point P across line Y. Point S is the reflection of point Z across line G. Point D is the reflection of point Z across point S. What is the length of segment UD? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let PD = 4. Point L is the reflection of point P across point D. Let Z be the midpoint of segment LD. Let M be the rotation of point P by angle 3pi/4 around point Z. Point T is the reflection of point L across point M. Line X is the perpendicular bisector of segment MD. Point B is the midpoint of segment TP. Let U be the reflection of line X through point L. Point W is the intersection of segment MZ and line U. Point R is the reflection of point W across segment BL. Find the measure of angle ZWR, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let IO = 2. Let A be the reflection of point O across point I. Line J passes through points A and O. Let F be the line through point A perpendicular to line J. Point H is on line F with |HI| = 1. Let Q be the angle bisector of angle HIO. Point S is the reflection of point O across point A. Let D be the reflection of line Q through point O. Point G is the reflection of point S across line D. Find the length of segment GS. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FW = 12. Line S is the perpendicular bisector of segment FW. Point Y is the reflection of point W across point F. Let A be the rotation of point W by angle pi/2 around point Y. Let Q be the midpoint of segment YA. Point O is the midpoint of segment AF. Line I passes through points A and F. Line G is the reflection of line S across point O. Let K be the reflection of line G across line I. Let B be the midpoint of segment OA. Let C be the reflection of point B across point Q. Point P is the reflection of point F across line K. Point M is the reflection of point B across line K. Point L is the reflection of point A across point C. Line H passes through point M perpendicular to line G. Point U is the reflection of point P across line H. Find the length of segment LU. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let KX = 11. Line Z passes through points X and K. Point F is obtained by rotating point X by angle 7pi/12 about point K. Let M be the line through points K and F. Line C is the reflection of line M across line Z. Let I be the reflection of point K across point X. Let L be the line through point F perpendicular to segment XF. Point V is the midpoint of segment XF. Point T is the reflection of point I across line C. Point D is the reflection of point V across line L. Find the distance between points T 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 ND = 9. Point Z is obtained by rotating point D by angle pi/3 about point N. Line A is the perpendicular bisector of segment NZ. Point S is the midpoint of segment DZ. Let J be the rotation of point N by angle 7pi/12 around point D. Point Q is the reflection of point S across line A. Let K be the perpendicular bisector of segment QJ. Let L be the midpoint of segment SQ. Point U lies on line K. Let W be the rotation of point L by angle 2pi/3 around point S. Point F is the reflection of point W across segment NZ. Point X is the reflection of point D across segment NZ. Point G is the midpoint of segment XU. Let V be the midpoint of segment FG. Point I is the midpoint of segment GN. What is the distance between points V and I? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WN = 1. Point V is obtained by rotating point W by angle pi/2 about point N. Let Y be the line through points N and V. Point R is the reflection of point W across line Y. Let X be the reflection of point V across point N. Q is placed 4 units away from R. Point F is the reflection of point Q across point N. Point T is on line Y with |TQ| = 2. Let S be the angle bisector of angle QTF. Point L is on line S with |LQ| = 7. Let I be the line through point T perpendicular to line S. Point K is the reflection of point L across line I. Line A is the perpendicular bisector of segment XK. Point E is the reflection of point R across line A. What is the length of segment EK? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FL = 1. Let G be the reflection of point F across point L. Let S be the reflection of point F across point G. Line P is the angle bisector of angle FLG. Point H is the reflection of point S across line P. Line T is the perpendicular bisector of segment GS. Let B be the rotation of point F by angle pi/12 around point H. Point U is the midpoint of segment HB. Point I is the reflection of point B across line T. Line Q passes through points G and U. Point V is the intersection of segment IL and line Q. What is 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]
MB = 10. Point A is the reflection of point B across point M. Let G be the reflection of point B across point A. Point N is the midpoint of segment MB. Let W be the angle bisector of angle GNB. Line R is the reflection of line W across point G. Point U is the reflection of point M across line R. What is the distance between points U 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 ZK = 2. Let B be the midpoint of segment ZK. Point V is obtained by rotating point B by angle 11pi/12 about point K. Let J be the line through points K and Z. Let T be the perpendicular bisector of segment VB. Let D be the midpoint of segment BK. Line I is the reflection of line T across line J. Point C is the reflection of point Z across point D. Let M be the reflection of point V across point C. Line O passes through points M and C. Line H is the reflection of line O across line J. Let W be the reflection of line O across line I. Lines W and H intersect at point F. Line H and segment MZ intersect at point N. What is the distance between points N and F? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let JS = 8. Point X is the midpoint of segment JS. Line O is the perpendicular bisector of segment SJ. Let V be the midpoint of segment JX. Point G is the midpoint of segment SV. Point Y lies on line O at distance 3 from point G. Point B is the midpoint of segment YJ. Let N be the midpoint of segment BS. Let Q be the reflection of point N across point S. Line K is the perpendicular bisector of segment QG. Point R is the reflection of point B across line K. Let P be the reflection of point B across point Y. Find the length of segment PR. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JR = 7. Line V passes through point R perpendicular to segment JR. Point P is the reflection of point R across point J. Let A be the reflection of line V through point J. Let I be the reflection of line A across line V. Point H is the reflection of point J across line I. Point Z is the midpoint of segment HR. Find the distance between points P and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let QA = 11. Let V be the vector from point A to point Q. Point D is the translation of point Q by vector V. Line U passes through points A and D. Let W be the reflection of point A across point D. Point K is obtained by rotating point W by angle pi/4 about point A. Line Z is the angle bisector of angle KQA. Point H is the reflection of point K across point D. Point B is the translation of point K by vector V. Let T be the line through point H perpendicular to line Z. Point F is the midpoint of segment HD. Let Y be the reflection of line U through point F. Point M is obtained by rotating point Q by angle 2pi/3 about point K. Point G is the midpoint of segment WB. Line O passes through point M perpendicular to line T. Line P is the reflection of line O across line Y. Point C is the reflection of point K across line P. Find the distance between points C and G. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FN = 7. Point G is the midpoint of segment FN. Point S is obtained by rotating point F by angle 7pi/12 about point G. Let Z be the rotation of point F by angle pi/4 around point S. Let E be the midpoint of segment ZS. Point W is the reflection of point F across point E. Point R is the midpoint of segment EG. What is the distance between points W 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 AC = 8. Let D be the rotation of point A by angle 7pi/12 around point C. Let F be the angle bisector of angle ADC. Let Z be the midpoint of segment DA. Let Q be the reflection of point C across point A. Let X be the line through point D parallel to segment AC. Line L is the reflection of line X across line F. Line E passes through point Z parallel to line L. Let V be the rotation of point Q by angle 11pi/12 around point A. Line O passes through point A perpendicular to segment AC. Point W is the reflection of point V across point C. Line N is the reflection of line O across line E. Lines X and N intersect at point H. Point I is the reflection of point W across segment AC. What is the distance between points I 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 PD = 8. Let Q be the perpendicular bisector of segment DP. Point I lies on line Q at distance 9 from point D. Point J is the reflection of point P across segment DI. Let Z be the midpoint of segment PI. Find the length of segment ZJ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WY = 8. Let I be the rotation of point Y by angle 2pi/3 around point W. Line S is the perpendicular bisector of segment WI. Let P be the line through point W parallel to line S. Line P and segment IY intersect at point A. Point X is the reflection of point A across segment YW. Find the distance between points A 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 IO = 5. Let Y be the rotation of point I by angle pi/12 around point O. S is placed 6 units away from O. Let Q be the angle bisector of angle SOI. Point Z is the reflection of point S across line Q. Let A be the rotation of point I by angle pi/3 around point Z. Point U is the reflection of point Y across segment ZI. Let M be the rotation of point O by angle 11pi/12 around point U. Let N be the midpoint of segment YM. Point K is the intersection of segments NO and UA. Find the length of segment IK. 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 LX = 10. Let Y be the reflection of point L across point X. Let A be the rotation of point L by angle 3pi/4 around point Y. Point R is the reflection of point A across point X. Line B is the angle bisector of angle RAL. Let G be the line through point L perpendicular to segment XA. Line I is the reflection of line B across point L. Line Z is the reflection of line I across line G. Lines Z and I intersect at point M. Find the length of segment RM. 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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PA = 7. Point Y is the midpoint of segment AP. Point N is the reflection of point A across point P. Let I be the angle bisector of angle NPA. Let E be the midpoint of segment NP. Line M is the perpendicular bisector of segment AP. Let L be the rotation of point A by angle pi/4 around point N. Line V is the reflection of line I across line M. Line O is the reflection of line V across point N. Point U is the reflection of point L across point E. Let B be the reflection of point U across point Y. Point J is the reflection of point Y across line O. What is the distance between points J and B? 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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TG = 2. Let Y be the vector from point G to point T. Point E is the midpoint of segment GT. Point D is the translation of point T by vector Y. Let S be the perpendicular bisector of segment ED. Point J is obtained by rotating point T by angle 5pi/6 about point G. Line X passes through point G parallel to line S. Point B is the midpoint of segment EJ. Point W is the reflection of point T across line S. Point F is the intersection of line X and segment BD. Find the distance between points W 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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Let TG = 6. Point N is obtained by rotating point G by angle 3pi/4 about point T. Let C be the rotation of point T by angle pi/6 around point N. Line W passes through points N and C. Point D is the reflection of point T across line W. Let H be the line through point G parallel to line W. Point L is the reflection of point D across line H. Find the length of segment LD. 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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PC = 8. Let A be the perpendicular bisector of segment PC. Point Y is obtained by rotating point P by angle pi/12 about point C. Let O be the line through point P parallel to line A. Line N passes through point Y parallel to line O. Point L is the reflection of point P across segment YC. Let W be the rotation of point L by angle 3pi/4 around point Y. Point H is the reflection of point C across point Y. Let G be the line through point W perpendicular to segment YC. Line R is the reflection of line O across line G. Lines N and R intersect at point D. Find the length of segment HD. 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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LY = 10. Let M be the perpendicular bisector of segment YL. Line J is the reflection of line M across point L. Line K is the reflection of line J across point Y. Point H is the reflection of point Y across line J. Point T is the midpoint of segment HL. Let I be the rotation of point H by angle 11pi/12 around point Y. Let S be the rotation of point T by angle 3pi/4 around point I. Point G is the intersection of segment LI and line K. Point A is the reflection of point G across line J. Let V be the midpoint of segment SI. Find the length of segment VA. 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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YA = 6. Let J be the midpoint of segment AY. Point S is obtained by rotating point Y by angle pi/6 about point J. Line X is the angle bisector of angle SYA. Point E is the reflection of point J across point A. Let T be the line through point S parallel to line X. Point R is the reflection of point E across point S. Point B is the reflection of point Y across line T. Find the distance between points B and R. Express your answer as a decimal number rounded to 2 decimal places.

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


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DN = 6. Point T is obtained by rotating point D by angle pi/3 about point N. Point C is the midpoint of segment DT. Line L passes through point N parallel to segment DT. Let M be the midpoint of segment CD. Let V be the midpoint of segment MT. Let U be the perpendicular bisector of segment DT. Point K is the reflection of point V across line U. Point I is the reflection of point V across line L. What is the length of segment KI? 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 XU = 4. Point R is the reflection of point X across point U. C is placed 7 units away from R. Point O is the reflection of point C across segment XU. Line G is the perpendicular bisector of segment OR. Let P be the midpoint of segment XU. Point J is the reflection of point P across line G. Find the distance between points J and O. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let FC = 11. Let Z be the midpoint of segment CF. Point M is obtained by rotating point C by angle 7pi/12 about point Z. Point D is obtained by rotating point Z by angle pi/6 about point M. Let L be the line through point M perpendicular to segment CM. Let Y be the vector from point M to point F. Let O be the reflection of point M across point Z. Point V is the translation of point O by vector Y. Point E is the translation of point V by vector Y. Line T is the perpendicular bisector of segment VE. Point G is the reflection of point C across line T. Point B is the midpoint of segment CM. Point H is the reflection of point E across line L. Let I be the reflection of point H across point D. Line S is the angle bisector of angle BCG. Point P is the reflection of point D across line S. Find the length of segment IP. Express your answer as a decimal number rounded to 2 decimal places.

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GD = 9. Point Q is obtained by rotating point G by angle 5pi/6 about point D. Let V be the perpendicular bisector of segment GD. Point H is obtained by rotating point G by angle 7pi/12 about point D. Line T passes through point G parallel to line V. Let U be the reflection of point D across point Q. Point J is the reflection of point H across line T. Let N be the reflection of line T across line V. Line F passes through point U perpendicular to line N. Point R is on line F. Let C be the reflection of point R across point J. Point E is the reflection of point C across line N. What is the length of segment RE? 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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VY = 1. Let M be the midpoint of segment YV. Let B be the perpendicular bisector of segment YV. Let F be the rotation of point Y by angle 7pi/12 around point V. Point S is the reflection of point M across point V. Let T be the midpoint of segment SV. Let P be the midpoint of segment FM. Point C is the reflection of point S across line B. Point N is the reflection of point C across segment FM. Let R be the midpoint of segment FP. Let J be the reflection of point R across point S. Let D be the line through point J perpendicular to segment CN. Point I is the reflection of point Y across line D. Find the length of segment IT. Express your answer as a decimal number rounded to 2 decimal places.

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


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TP = 2. Point X is the reflection of point P across point T. Let W be the rotation of point T by angle pi/3 around point P. Point K is the reflection of point W across point T. Let L be the midpoint of segment XK. Line O passes through points T and P. Point V lies on segment KL. Point N is the reflection of point V across line O. What is the measure of angle NVK, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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TG = 8. Let W be the midpoint of segment GT. X is constructed so that GX = 5. B is placed 5 units away from W. Point N is obtained by rotating point T by angle 5pi/12 about point X. Line Q is the angle bisector of angle WXN. Let J be the midpoint of segment WB. Let K be the line through point J perpendicular to line Q. Point M is the reflection of point W across line K. 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}


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Let IM = 2. Point A is the reflection of point M across point I. Line F passes through points A and I. Point O lies on line F. Line C is the perpendicular bisector of segment MI. Let P be the reflection of point M across point O. Let X be the perpendicular bisector of segment OM. Point G is the reflection of point O across line C. Point Z is the reflection of point G across line X. Find the distance between points P and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RW = 9. Let Y be the rotation of point W by angle 11pi/12 around point R. Line Q is the perpendicular bisector of segment YR. Let U be the rotation of point W by angle pi/2 around point R. Let X be the line through point W perpendicular to line Q. Point A is obtained by rotating point Y by angle 3pi/4 about point U. Point V is obtained by rotating point Y by angle 5pi/12 about point W. Point Z is the reflection of point R across line X. Point J is the reflection of point Y across segment VZ. Let L be the angle bisector of angle AWV. Point N is the reflection of point W across segment VZ. Let D be the line through point J perpendicular to segment VZ. Point T is the reflection of point N across line D. Point C is the reflection of point T across line L. What is the length of segment CJ? 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 HE = 11. Point U is the midpoint of segment HE. Point J is obtained by rotating point U by angle 2pi/3 about point E. Let W be the angle bisector of angle HJU. Let O be the line through point E parallel to line W. Let D be the reflection of point J across point H. Point L is on line O with |LU| = 5. Point R is the reflection of point U across line O. Let Y be the line through point D parallel to line O. Line X is the reflection of line Y across point L. Point M is the reflection of point D across line X. Let T be the midpoint of segment MR. What is the length of segment MT? 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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NU = 10. Point Z is obtained by rotating point N by angle 3pi/4 about point U. Let R be the reflection of point N across point U. Point L is obtained by rotating point Z by angle pi/2 about point R. Line H passes through point L parallel to segment NR. Let P be the midpoint of segment ZL. Point X is the reflection of point L across segment NR. Point C is the reflection of point P across line H. What is the measure of angle CLX, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let NM = 10. Point W is the reflection of point N across point M. Let Z be the rotation of point M by angle 2pi/3 around point W. Let K be the reflection of point Z across point N. Line X passes through points Z and N. Point A is the reflection of point K across point Z. Let P be the line through point A perpendicular to line X. Point L is the reflection of point Z across line P. Let O be the reflection of point L across point M. 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]
BY = 11. Point M is obtained by rotating point B by angle pi/4 about point Y. Point C is the midpoint of segment MB. Point E is obtained by rotating point Y by angle 11pi/12 about point C. Let J be the reflection of point Y across point B. Line U passes through point J perpendicular to segment YB. Point R is the reflection of point E across line U. Let Q be the reflection of point R across point E. What is the length of segment QC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let PF = 12. Point W is obtained by rotating point F by angle 5pi/6 about point P. Let A be the rotation of point P by angle pi/6 around point F. Let V be the midpoint of segment FP. Point B is the midpoint of segment AW. Line K is the angle bisector of angle PAB. Let O be the perpendicular bisector of segment VF. Let R be the reflection of line K through point V. Let U be the line through point F parallel to line O. Point M is the intersection of lines R and U. Find the distance between points M 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 LK = 12. Line G passes through points K and L. Let Z be the reflection of point K across point L. Let X be the rotation of point K by angle 2pi/3 around point L. Line P passes through point L perpendicular to line G. Point V is the midpoint of segment XL. Line A is the reflection of line P across point K. Point C is obtained by rotating point V by angle 5pi/12 about point L. Line N is the perpendicular bisector of segment KC. Line R passes through point Z parallel to line N. Line T is the reflection of line R across line A. Lines G and T intersect at point J. What is the distance between points J and C? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HU = 7. Line F is the perpendicular bisector of segment HU. Point T lies on line F at distance 9 from point U. Let K be the reflection of point T across point U. Let S be the rotation of point K by angle 5pi/12 around point T. Let P be the midpoint of segment UT. Let O be the midpoint of segment SU. Point D is the reflection of point P across point O. Let G be the reflection of point T across point D. Point W is the reflection of point O across point U. What is the distance between points W 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 SQ = 9. Let C be the reflection of point Q across point S. Let N be the rotation of point C by angle 3pi/4 around point S. Let V be the midpoint of segment QS. Line K is the perpendicular bisector of segment CQ. Point R is the reflection of point N across line K. Point B is the reflection of point N across point Q. Let L be the reflection of point V across point C. Point Y is the reflection of point B across segment NV. Let T be the line through points Y and S. Line G passes through point R perpendicular to line T. Point J is the reflection of point L across line G. Point P is the midpoint of segment NV. Find the distance between points P and J. Express your answer as a decimal number rounded to 2 decimal places.

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


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ZA = 3. Let H be the reflection of point A across point Z. Point O is obtained by rotating point A by angle 2pi/3 about point H. Point T is the reflection of point Z across point H. Point X is the midpoint of segment HZ. Let P be the line through point X perpendicular to segment XO. Line L is the reflection of line P across point T. Line L and segment XO intersect at point F. Find the distance between points O and F. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
UD = 3. Let C be the reflection of point D across point U. Line F is the perpendicular bisector of segment UC. Point H is the reflection of point D across line F. Line N is the angle bisector of angle DCH. Let Q be the reflection of line N through point U. Point W is the reflection of point C across line Q. 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}


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TX = 2. Point L is the midpoint of segment TX. Point E is the midpoint of segment XL. Let A be the midpoint of segment EX. What is the distance between points L 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 LE = 3. Let Q be the line through point L perpendicular to segment EL. Point B is on line Q. Point T is the midpoint of segment EL. Point R is the reflection of point B across segment EL. Let A be the midpoint of segment TR. Point C is the reflection of point A across line Q. What is the length of segment CA? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QP = 7. Point C is the midpoint of segment PQ. Let K be the rotation of point P by angle 11pi/12 around point C. Point X is the midpoint of segment QK. Let D be the reflection of point K across point P. Point N is the midpoint of segment PD. Let Y be the midpoint of segment XN. Let Z be the reflection of point C across point D. Point R is the reflection of point Y across segment QZ. Find the distance between points C 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 CG = 9. Let Q be the rotation of point C by angle pi/12 around point G. Point L is obtained by rotating point C by angle 2pi/3 about point G. Let D be the rotation of point L by angle 5pi/6 around point Q. Point H is the reflection of point D across segment QC. Point S is the reflection of point H across segment QL. Point Z is the midpoint of segment QG. Let U be the reflection of point D across point Z. Find the distance between points S 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 DX = 8. Let B be the midpoint of segment XD. Let Q be the perpendicular bisector of segment BD. Point V is obtained by rotating point B by angle pi/6 about point X. Let P be the rotation of point X by angle 11pi/12 around point D. Line W is the perpendicular bisector of segment PB. Let N be the reflection of line Q across line W. Let A be the midpoint of segment PB. Let S be the line through point B parallel to line Q. Point F is the reflection of point V across line Q. Point U is the reflection of point A across line S. Segment PB and line N intersect at point Z. Point O is the midpoint of segment FZ. Find the distance between points O 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 PB = 3. Line H is the perpendicular bisector of segment BP. Point O is obtained by rotating point B by angle 3pi/4 about point P. Line Z is the angle bisector of angle POB. Let Y be the rotation of point P by angle 7pi/12 around point O. Line F is the reflection of line Z across line H. Point V is on line F with |VP| = 7. Line K passes through point B parallel to line F. Let U be the line through points P and V. Point N is the intersection of lines U and K. Line G passes through point Y parallel to line K. Point W is the reflection of point V across line G. Find the distance between points N and W. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SX = 3. Let B be the midpoint of segment SX. Let C be the angle bisector of angle SBX. Line F passes through point X parallel to line C. Point Z lies on line F at distance 11 from point B. Point U is the midpoint of segment BX. Line H is the perpendicular bisector of segment UX. Let I be the reflection of line H across line F. Point P is the midpoint of segment ZS. Point G is the reflection of point Z across line I. Let M be the midpoint of segment PU. Find the distance between points G and M. 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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QV = 4. Let H be the midpoint of segment VQ. Let Y be the rotation of point Q by angle 11pi/12 around point H. Line C is the angle bisector of angle QYV. Let A be the line through point V parallel to line C. Point G is the reflection of point H across line A. Point W is the midpoint of segment YQ. Find the length of segment GW. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let AN = 10. Let E be the rotation of point N by angle pi/2 around point A. Let B be the line through point A perpendicular to segment AN. Let S be the reflection of point A across point E. Line K is the perpendicular bisector of segment AN. Line D is the reflection of line B across line K. Point X is the reflection of point E across line D. Point M is the reflection of point X across line K. Point H is the reflection of point M across point S. What is the length of segment AH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VY = 2. Point D is obtained by rotating point Y by angle 11pi/12 about point V. Line G is the perpendicular bisector of segment YV. Line A passes through point V parallel to line G. Point X is obtained by rotating point V by angle 5pi/6 about point D. Let S be the reflection of line A across line G. Let Z be the perpendicular bisector of segment YD. Line P is the reflection of line G across line Z. Point U is the reflection of point V across point X. Point N is the reflection of point U across line P. Let T be the line through point N parallel to segment YX. Line A and segment YX intersect at point C. Lines T and S intersect at point K. Let M be the reflection of point K across point C. Point O is the reflection of point K across point U. Let Q be the midpoint of segment OM. Find the length of segment KQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let FC = 12. Point A is obtained by rotating point C by angle 5pi/6 about point F. Let X be the angle bisector of angle ACF. Point K is the reflection of point F across line X. Point I is the reflection of point F across point A. Point O is the midpoint of segment KC. Point U is obtained by rotating point F by angle pi/2 about point I. Let B be the line through points I and U. Point L is the midpoint of segment OC. Let Z be the midpoint of segment UA. Line N is the perpendicular bisector of segment ZI. Point D is the reflection of point K across line N. Point R is the reflection of point L across line B. Let Y be the midpoint of segment DL. Find the length of segment RY. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let MT = 11. Point L is obtained by rotating point T by angle 5pi/12 about point M. Let U be the vector from point M to point L. Let P be the rotation of point L by angle pi/2 around point T. Point N is the reflection of point T across point M. Line X is the angle bisector of angle TNP. Point G is the translation of point P by vector U. Line R is the perpendicular bisector of segment LG. Point Y is obtained by rotating point T by angle 5pi/6 about point L. Point J is the reflection of point P across line R. Point H is the midpoint of segment NG. Let Q be the line through point L perpendicular to line X. Point B is the reflection of point G across line Q. Point O is the reflection of point M across point P. Point V is the midpoint of segment HB. Point W is the midpoint of segment YJ. Let Z be the line through points V and W. Point A is the reflection of point O across line Z. Point E is the midpoint of segment WA. What is the length of segment OE? 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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VS = 7. Let M be the line through points S and V. Point F is on line M. Let U be the midpoint of segment VS. Let Y be the rotation of point U by angle pi/4 around point V. Line O is the angle bisector of angle FUS. Point J is the reflection of point Y across line O. Point L is the reflection of point S across point J. Let B be the midpoint of segment LJ. Point R is the reflection of point B across point J. Let X be the reflection of point R across point B. What is the distance between points V and X? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IN = 2. Let D be the perpendicular bisector of segment NI. Line L is the reflection of line D across point I. Point Z is the reflection of point N across point I. Let K be the midpoint of segment IN. Point Q is the reflection of point N across line L. Let J be the rotation of point K by angle 5pi/6 around point N. Let W be the rotation of point Q by angle 3pi/4 around point K. Line M is the perpendicular bisector of segment QJ. Point Y is the intersection of line L and segment WZ. Line T passes through points N and Y. Let C be the reflection of line T across line M. Point A is the reflection of point J across line C. Point O is the midpoint of segment AI. Find the distance between points O and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GN = 8. Point Y is the reflection of point N across point G. Let A be the perpendicular bisector of segment NG. Let R be the reflection of line A through point Y. Point S is the reflection of point Y across line R. Let Q be the midpoint of segment SG. Let J be the perpendicular bisector of segment SN. Let P be the rotation of point N by angle 2pi/3 around point G. Point X is the reflection of point G across line J. Point O is the reflection of point P across line J. Point F is the reflection of point P across segment NG. Point D is the reflection of point G across point O. Point I is the reflection of point Q across segment XP. Let K be the midpoint of segment ID. What is the length of segment FK? 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 WP = 3. Point C is obtained by rotating point P by angle pi/12 about point W. Point R is the midpoint of segment CP. Let M be the midpoint of segment WP. Point K is the reflection of point R across segment CW. What is the length of segment MK? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EQ = 2. Point F is obtained by rotating point Q by angle pi/6 about point E. Let K be the rotation of point E by angle pi around point F. Point U is the reflection of point K across point E. Line C is the angle bisector of angle UQK. Let M be the reflection of point K across point U. Point X is the reflection of point M across line C. Find the distance between points X and U. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GQ = 11. Point M is the reflection of point Q across point G. Point Y is obtained by rotating point M by angle pi/12 about point Q. Point U is the reflection of point Y across point G. Point A is obtained by rotating point M by angle pi/3 about point U. Point Z is the reflection of point G across point A. Point X is the midpoint of segment ZU. Point H is the reflection of point A across segment ZU. What is the length of segment HX? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EC = 6. Point V is the midpoint of segment EC. Point J is the reflection of point V across point E. Let D be the midpoint of segment JC. Let H be the perpendicular bisector of segment JD. Point B is the reflection of point E across line H. What is the distance between points B and E? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GC = 7. Let Y be the midpoint of segment CG. Line F is the angle bisector of angle CGY. Let K be the line through point Y perpendicular to line F. Line X passes through point G parallel to line K. Let M be the reflection of point G across point C. Point D is on line X with |DM| = 3. Point A is the reflection of point D across line F. Point R is the midpoint of segment MA. Let V be the angle bisector of angle GAR. Point J is the reflection of point M across line V. Let Z be the midpoint of segment JC. Find the distance between points Z and J. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BR = 5. Line Y passes through points B and R. Point A is the midpoint of segment RB. Let F be the rotation of point R by angle 5pi/12 around point A. Point L is the reflection of point F across line Y. Line M passes through point L parallel to line Y. Let G be the line through point F parallel to line Y. Point W is on line M. Point K is the reflection of point W across line G. What is the length of segment WK? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
TG = 8. Point I is the reflection of point G across point T. Let Z be the rotation of point I by angle 7pi/12 around point G. Let R be the line through points I and Z. Point N is the midpoint of segment ZG. Let K be the midpoint of segment IN. Point O is the reflection of point K across line R. Let A be the reflection of point N across point G. Let S be the midpoint of segment KA. Let H be the midpoint of segment OG. Point P is the reflection of point H across point N. Point U is the reflection of point S across point N. Find the distance between points U and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GM = 12. Let U be the perpendicular bisector of segment GM. Line K passes through point M parallel to line U. Point J lies on line K at distance 8 from point G. Point A lies on line K at distance 6 from point G. Let V be the perpendicular bisector of segment AM. Let H be the midpoint of segment AJ. Point Z is the reflection of point H across line V. Point L is the midpoint of segment AG. Find the length of segment ZL. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
DX = 9. Let B be the vector from point X to point D. Let I be the midpoint of segment DX. Point F is obtained by rotating point X by angle 7pi/12 about point D. Let L be the midpoint of segment FD. Point Y is the midpoint of segment LX. Point Z is the translation of point I by vector B. Let J be the midpoint of segment ZL. Let P be the midpoint of segment JY. Point W is the reflection of point Z across point P. Point A is the reflection of point P across segment FL. Point G is the midpoint of segment WA. Point H is the midpoint of segment GI. What is the length of segment HJ? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
KX = 10. Point B is the reflection of point X across point K. Let R be the reflection of point X across point B. Let D be the vector from point B to point K. Let E be point R translated by vector D. Point U is obtained by rotating point K by angle pi/6 about point X. Let M be the midpoint of segment EU. Point O is the reflection of point B across point U. Point N is the reflection of point K across point M. Point W is the reflection of point N across segment KE. What is the distance between points W and O? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let TC = 2. Point P is the reflection of point T across point C. Let V be the vector from point T to point C. Let B be the rotation of point P by angle pi/6 around point T. Let W be the angle bisector of angle CBP. Point G is the reflection of point C across point P. Point U is the reflection of point B across point C. Point M is obtained by rotating point B by angle 11pi/12 about point U. Let E be the line through point T parallel to line W. Point H is the translation of point M by vector V. Let Z be the line through points T and P. Point F is obtained by rotating point H by angle 5pi/12 about point C. Point O is the midpoint of segment MP. Point D is the reflection of point G across point F. Let N be the reflection of line Z across line E. Point K is the reflection of point O across line N. Find the length of segment DK. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let OM = 5. Let B be the reflection of point O across point M. Point Q is obtained by rotating point O by angle 7pi/12 about point M. Line H is the perpendicular bisector of segment OQ. Point C is the midpoint of segment BM. Let Z be the perpendicular bisector of segment BQ. Let W be the line through point O perpendicular to line Z. Point X is the reflection of point C across point Q. Point D is the reflection of point X across line W. Let S be the reflection of point C across point D. Point A is the reflection of point C across line H. Find the distance between points A and S. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
KD = 7. Let G be the midpoint of segment KD. Let I be the rotation of point G by angle pi/2 around point K. Point Y is the reflection of point I across point G. Let E be the line through points I and D. Let U be the line through point D perpendicular to line E. Let L be the reflection of line U through point I. Point Z is the reflection of point Y across line L. Find the distance between points I and Z. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let IP = 3. Point R is the reflection of point I across point P. Point O is obtained by rotating point R by angle pi/3 about point I. Point H is obtained by rotating point P by angle pi/4 about point I. Line Q is the angle bisector of angle HIP. Line Y passes through point R parallel to line Q. Point N is obtained by rotating point H by angle pi about point P. Point Z is the midpoint of segment ON. Let V be the perpendicular bisector of segment IH. Let W be the line through point N perpendicular to line Y. Point T is the reflection of point I across point Z. Line X is the reflection of line V across point R. Point S is the midpoint of segment PT. Point L is the reflection of point N across line X. Point K is the reflection of point S across line W. What is the length of segment KL? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let QC = 7. Line F passes through point Q perpendicular to segment QC. Point N lies on line F at distance 6 from point Q. Point K is the reflection of point C across point N. Let X be the midpoint of segment QC. Let Z be the midpoint of segment XQ. Point V is the reflection of point K across segment QC. Point E is the reflection of point Z across segment QK. What is the length of segment EV? Express your answer as a decimal number rounded to 2 decimal places.

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