[User]
RD = 5. Line F is the perpendicular bisector of segment DR. Point B is on line F with |BD| = 1. Line Y is the perpendicular bisector of segment BD. Point Q is the midpoint of segment BR. Point U is the reflection of point R across line Y. Let N be the midpoint of segment BQ. Point V is the midpoint of segment DR. Line J is the reflection of line Y across point N. Point P is the reflection of point Q across line J. Point S is the intersection of segment PV and line J. What is 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}


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Let XY = 3. Let G be the perpendicular bisector of segment XY. Point U is obtained by rotating point X by angle 5pi/6 about point Y. Let F be the midpoint of segment XU. Point A is the reflection of point U across line G. Let T be the vector from point F to point X. Let W be point X translated by vector T. Point E is the reflection of point X across point W. Point P is the translation of point A by vector T. Point J is the reflection of point F across line G. Let Z be the line through points U and P. Let V be the line through point E parallel to segment JY. Point M is the intersection of lines V and Z. What is the length of segment WM? 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 DU = 4. Point B is the midpoint of segment UD. Point S is the midpoint of segment BD. Point G is the reflection of point S across point U. Let W be the rotation of point S by angle 7pi/12 around point G. Point K is the midpoint of segment GW. Let O be the rotation of point D by angle 11pi/12 around point U. Point E is obtained by rotating point K by angle 7pi/12 about point O. Let N be the rotation of point O by angle pi/6 around point U. Let Y be the reflection of point U across point N. Point T is the midpoint of segment OG. Let A be the angle bisector of angle YET. Point H is the reflection of point B across line A. What is the length of segment NH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let IY = 3. Point D is the midpoint of segment YI. Point Z is obtained by rotating point D by angle pi/12 about point Y. Point X is obtained by rotating point Z by angle 3pi/4 about point D. Let K be the midpoint of segment XD. Let Q be the midpoint of segment KI. Point J is the reflection of point X across segment KI. What is the distance between points Q and J? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QE = 9. Point L is the reflection of point Q across point E. Let Y be the midpoint of segment QE. Point R is the reflection of point Y across point L. Let O be the reflection of point Q across point R. Point I is the reflection of point E across point O. Point P is the reflection of point E across point R. Find the distance between points P 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 TJ = 8. Point X is the midpoint of segment TJ. Let Q be the rotation of point X by angle 11pi/12 around point J. Let L be the angle bisector of angle XJT. Line P passes through point Q perpendicular to line L. Point C is the reflection of point X across line P. Point F is obtained by rotating point C by angle 7pi/12 about point Q. Let N be the line through point C parallel to line P. Let H be the reflection of point T across point J. Point O is the reflection of point F across line N. Point R is the midpoint of segment OH. What is the length of segment FR? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FT = 3. Let M be the reflection of point F across point T. Point A is the midpoint of segment TM. Line K is the perpendicular bisector of segment FM. Point C is the reflection of point F across point A. Line P is the angle bisector of angle CMF. Point D is the reflection of point A across line K. Line G is the perpendicular bisector of segment AF. Point B is obtained by rotating point M by angle 5pi/6 about point D. Line U is the reflection of line G across line K. Let W be the reflection of point B across point F. Point Z is the reflection of point W across line U. Point S is the reflection of point D across line P. What is the length of segment SZ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GT = 1. Let P be the rotation of point T by angle 3pi/4 around point G. Let I be the reflection of point T across point G. Line L passes through point I perpendicular to segment GP. Point F is the reflection of point T across line L. Line C passes through point T perpendicular to segment GP. Let D be the perpendicular bisector of segment IF. Line W is the reflection of line D across line C. Point U is the reflection of point F across line W. What is the distance between points P 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 ZC = 7. Let R be the vector from point Z to point C. Let L be the midpoint of segment ZC. Point M is the reflection of point L across point C. Point P is the reflection of point C across point M. Let Y be the midpoint of segment PM. Let B be point Y translated by vector R. Let D be the perpendicular bisector of segment ZY. Line U passes through point B perpendicular to line D. Point V is obtained by rotating point L by angle 11pi/12 about point P. Point I is the midpoint of segment YV. Let F be the line through point M perpendicular to line U. Point H is the reflection of point B across point Y. Point K is the reflection of point H across line F. 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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IL = 4. Let T be the rotation of point I by angle 7pi/12 around point L. Point W is the reflection of point L across point I. Let P be the midpoint of segment IW. Let V be the line through points T and P. Point Y is the reflection of point I across line V. What is the distance between points Y and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PA = 1. Let N be the reflection of point P across point A. Line F is the perpendicular bisector of segment NP. Let O be the reflection of point N across point P. Let I be the reflection of point P across point O. Line V is the perpendicular bisector of segment PO. Let Q be the line through point I parallel to line V. Let J be the rotation of point I by angle pi/3 around point O. Let W be the reflection of line F across line Q. Let B be the midpoint of segment AI. Point L is obtained by rotating point N by angle pi/2 about point O. Line H passes through points J and L. Lines W and H intersect at point D. Point M is the reflection of point P across segment BL. Point T is the midpoint of segment DM. What is 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 MP = 6. Let W be the vector from point P to point M. Let A be the line through points P and M. Line F passes through point P perpendicular to line A. Line E passes through point M parallel to line F. Let L be the reflection of point P across point M. Line I is the reflection of line E across line F. Let Y be point L translated by vector W. Let G be the reflection of line F across line I. Point V is the midpoint of segment PY. Point R is on line G with |RV| = 2. Let D be the midpoint of segment LR. Point O is the midpoint of segment YD. Point J is the reflection of point O across segment LR. What is the length of segment YJ? 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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DV = 12. Let A be the rotation of point V by angle 11pi/12 around point D. Let R be the rotation of point V by angle pi/2 around point A. Point Z is the midpoint of segment VD. Point T is the midpoint of segment AR. Point N is the midpoint of segment VT. Let O be the midpoint of segment ZN. Point M is the reflection of point O across segment RV. Find the length of segment MO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SL = 6. Point R is obtained by rotating point S by angle 5pi/6 about point L. Let G be the perpendicular bisector of segment SR. Point M is the midpoint of segment SL. Line V passes through point M parallel to line G. Point H is the reflection of point L across line V. Find the distance between points R and H. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RE = 9. Let S be the line through points E and R. Let T be the vector from point R to point E. Let L be point E translated by vector T. Point Y is obtained by rotating point R by angle 5pi/6 about point L. Let B be the reflection of point Y across point L. Point P is the reflection of point B across point Y. Let W be the midpoint of segment LE. Point A is the midpoint of segment PW. Point K is the reflection of point A across line S. Point F is the reflection of point A across point K. Let V be the midpoint of segment FB. Let M be the midpoint of segment VB. What is the distance between points M 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 AM = 5. Let F be the midpoint of segment MA. Let P be the rotation of point A by angle 5pi/6 around point M. Line W is the perpendicular bisector of segment AF. Line I passes through points M and F. Line L passes through point P parallel to line W. Point R is the reflection of point M across line L. Let Q be the reflection of line W through point F. Lines Q and I intersect at point D. Let K be the midpoint of segment DR. Find the distance between points R and K. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BV = 7. Let T be the reflection of point B across point V. Let P be the line through points V and T. Line N passes through point T perpendicular to line P. Point Q is the reflection of point B across line N. 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]
Let QV = 6. Let B be the reflection of point Q across point V. Point E is obtained by rotating point Q by angle pi about point B. Let U be the midpoint of segment BE. Point T is the reflection of point E across point Q. Let A be the rotation of point U by angle 7pi/12 around point E. Line Y is the perpendicular bisector of segment TU. Let C be the perpendicular bisector of segment EQ. Let D be the reflection of line Y across line C. Point P is the reflection of point A across line D. Point L is the reflection of point A across point U. Find the length of segment LP. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OK = 5. Let P be the perpendicular bisector of segment OK. Line C is the reflection of line P across point O. Let G be the line through point O parallel to line C. Let H be the line through point K parallel to line P. Let W be the vector from point O to point K. Point F is obtained by rotating point K by angle pi/6 about point O. Point I is the midpoint of segment OK. Point B is the reflection of point F across point I. Let M be point K translated by vector W. Line N passes through points K and B. Line Y is the reflection of line H across line G. Lines Y and N intersect at point L. Let V be the reflection of point L across point I. Point Q is the reflection of point V across point M. Find the distance between points L and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YH = 6. Let J be the vector from point H to point Y. Point F is the translation of point Y by vector J. Point D is obtained by rotating point H by angle 7pi/12 about point F. Let W be point D translated by vector J. Point Z is obtained by rotating point F by angle 11pi/12 about point Y. Let S be the vector from point F to point Z. Point N is the translation of point Y by vector S. Let P be the angle bisector of angle HDY. Point R is the reflection of point F across line P. Point Q is the midpoint of segment RW. Point G is the reflection of point Q across segment NW. Point B is the midpoint of segment NW. Find the distance between points G and B. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NP = 3. Let R be the vector from point N to point P. Let Y be point P translated by vector R. Line X is the perpendicular bisector of segment NP. Let Q be the line through point Y parallel to line X. Point O is the midpoint of segment PN. Let J be the perpendicular bisector of segment PY. Point V is the reflection of point O across line Q. Point I lies on line J at distance 3 from point V. Let W be the reflection of line Q through point P. Let U be the line through point I parallel to segment PN. Line M is the reflection of line U across point N. Lines M and W intersect at point A. Find the length of segment YA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let IG = 1. Let C be the perpendicular bisector of segment GI. Let A be the line through point G parallel to line C. Let K be the midpoint of segment IG. Point H is the reflection of point K across point I. Point Y is the reflection of point H across line A. Point L is obtained by rotating point H by angle pi/2 about point G. Point E is the reflection of point Y across segment LH. Let T be the reflection of point G across point E. Point Q is the reflection of point K across segment LH. Find 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]
YW = 4. Let K be the vector from point Y to point W. Let P be point W translated by vector K. Point N is obtained by rotating point Y by angle 3pi/4 about point W. Line D is the perpendicular bisector of segment YP. Point U is the reflection of point Y across point N. Point R is the reflection of point N across line D. Point Z is the midpoint of segment PR. Point T is the reflection of point U across line D. What is the measure of angle TNZ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KY = 3. Point C is obtained by rotating point K by angle pi/6 about point Y. Point Q is the reflection of point Y across point C. Point D is the reflection of point Y across point Q. Let T be the reflection of point K across point Q. Let W be the midpoint of segment DK. What is the distance between points T and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let JT = 4. Point R is obtained by rotating point T by angle 5pi/12 about point J. Line X passes through points T and J. Line O passes through point R parallel to line X. Let A be the midpoint of segment TJ. Let L be the midpoint of segment JR. Point D is the midpoint of segment LT. Let W be the perpendicular bisector of segment DA. Let K be the line through point D perpendicular to line O. Line Y is the reflection of line O across line W. Line P is the reflection of line K across line Y. Point S is the reflection of point L across line P. Find the length of segment AS. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
CZ = 6. Let X be the midpoint of segment CZ. Point J is the midpoint of segment XC. Let G be the midpoint of segment XJ. Let M be the rotation of point Z by angle pi/12 around point J. Let Q be the midpoint of segment CG. Let D be the midpoint of segment GJ. Let K be the midpoint of segment DZ. Point F is the reflection of point Q across segment MX. Let B be the reflection of point J across point K. What is the distance between points F and B? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KP = 1. Point W is obtained by rotating point K by angle pi/6 about point P. Let L be the rotation of point K by angle 5pi/12 around point W. Let S be the angle bisector of angle PWL. Point X lies on line S at distance 8 from point W. Let N be the reflection of point L across point X. Point Q is the reflection of point L across line S. Point Z is the reflection of point K across segment NQ. Find the measure of angle ZPQ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HD = 5. Point T is obtained by rotating point H by angle pi/6 about point D. Let A be the line through points T and D. Let R be the reflection of point T across point H. Let P be the line through point D perpendicular to line A. Let M be the rotation of point H by angle pi/12 around point D. Point X is the midpoint of segment DR. Point B is the reflection of point X across line P. Line N passes through points X and T. Line Y is the angle bisector of angle XMB. Point U is the reflection of point T across line Y. Point E is the reflection of point H across line N. What is the distance between points E and U? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let IX = 10. Let B be the rotation of point I by angle 3pi/4 around point X. Point Q is obtained by rotating point X by angle pi/3 about point I. Line T is the angle bisector of angle QXB. Point P is the reflection of point I across line T. Point N is the reflection of point X across point B. Let W be the midpoint of segment PI. Let O be the reflection of point W across point N. Point J is the reflection of point O across segment PI. Find the distance between points I and J. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let JA = 1. Let Y be the rotation of point J by angle pi/4 around point A. Let L be the rotation of point A by angle pi/3 around point J. Point E is obtained by rotating point A by angle pi/6 about point Y. Line Q is the angle bisector of angle EJY. Let R be the rotation of point Y by angle 11pi/12 around point A. Let N be the midpoint of segment LR. Point G is the reflection of point E across point N. Line M passes through point L parallel to line Q. Point X is the reflection of point R across line M. Find the distance between points X and G. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WZ = 9. Point D is the reflection of point W across point Z. Line Y is the perpendicular bisector of segment ZW. Point A is the midpoint of segment DZ. Let U be the vector from point D to point W. Point T is the reflection of point D across line Y. Point G is the translation of point T by vector U. Point M is the reflection of point A across point W. Let I be the rotation of point G by angle pi/2 around point Z. Point X is obtained by rotating point T by angle 5pi/6 about point M. Point K is the midpoint of segment XI. Let L be the midpoint of segment KD. Point J is the reflection of point D across segment IK. What is the length of segment JL? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let EQ = 10. Let H be the vector from point Q to point E. Let L be point E translated by vector H. Line B is the perpendicular bisector of segment LQ. Point U is on line B with |UE| = 12. Let R be the angle bisector of angle QLE. Point F is the midpoint of segment UL. Line X is the reflection of line R across point F. Point W is the reflection of point Q across line X. Let T be the reflection of point W across point F. What is the distance between points E 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 ZY = 3. Let K be the perpendicular bisector of segment YZ. Point P is on line K. Let T be the rotation of point Z by angle 3pi/4 around point Y. Point F is the reflection of point T across line K. Let L be the midpoint of segment PZ. Line G is the reflection of line K across point L. Point O is the reflection of point T across line G. Find the length of segment OF. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
CA = 7. Line G is the perpendicular bisector of segment CA. Line I is the reflection of line G across point C. Point P is the reflection of point A across line I. Point F is the reflection of point P across line G. Point Y is the reflection of point C across line I. Point M is the midpoint of segment AF. What is 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 SP = 9. Point G is obtained by rotating point P by angle pi/3 about point S. Point U is the midpoint of segment SG. Line W is the angle bisector of angle GPU. Point V is the reflection of point U across line W. Let J be the reflection of point S across point P. Point L is the reflection of point V across segment SP. Line E is the perpendicular bisector of segment SP. Let M be the line through point J perpendicular to segment SP. Point I is the reflection of point U across line M. Let F be the midpoint of segment IP. Let K be the line through point L parallel to line W. Let H be the reflection of line E across line K. Point R is the reflection of point L across line H. Point Z is the reflection of point F across point R. Find the distance between points V and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QL = 6. Point S is obtained by rotating point Q by angle pi/4 about point L. Let E be the midpoint of segment SL. Line I is the angle bisector of angle LQE. Point V lies on segment EQ. Point X is the reflection of point L across segment VS. Point A is the reflection of point S across line I. Point U is the midpoint of segment SA. Point K is the midpoint of segment XA. What is the length of segment KU? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ON = 1. Let I be the perpendicular bisector of segment ON. Point T is on line I with |TN| = 8. Let S be the perpendicular bisector of segment NT. Line C passes through point T parallel to line S. Point Y is the reflection of point O across line C. Let X be the reflection of point N across point T. What is the distance between points X and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let YZ = 8. Let A be the rotation of point Z by angle pi around point Y. Let Q be the reflection of point Y across point Z. Let S be the rotation of point A by angle pi/4 around point Y. Let T be the midpoint of segment AS. Point G is the reflection of point A across point Z. Point D is the reflection of point T across point G. Line R is the angle bisector of angle GQS. Point J is the reflection of point D across line R. Line P is the perpendicular bisector of segment JQ. Point B is the reflection of point Y across line P. What is the distance between points B and J? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RX = 5. Let M be the line through points X and R. Let E be the rotation of point X by angle 7pi/12 around point R. Let A be the reflection of point X across point E. Let J be the line through point E perpendicular to line M. Let Z be the midpoint of segment XE. Line V passes through point Z perpendicular to line J. Line L passes through point A parallel to line V. Point I is the reflection of point Z across line L. Let D be the line through points E and I. Lines L and D intersect at point U. Find the distance between points I and U. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GY = 1. Line V is the perpendicular bisector of segment YG. Let U be the reflection of line V through point Y. Point F is the reflection of point Y across line U. Line S passes through points G and F. Point M is the midpoint of segment YF. Let N be the vector from point M to point G. Point A lies on line V at distance 12 from point F. Point E is the reflection of point A across line S. Let H be point E translated by vector N. Point R is the midpoint of segment EY. Let Q be the reflection of point H across point M. Point T is the midpoint of segment GR. Point P is the reflection of point A across point Q. Find the distance between points P and T. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
CA = 5. Point H is the reflection of point A across point C. Line D passes through points C and H. Line V passes through point A perpendicular to line D. Let L be the rotation of point H by angle pi/2 around point C. Point P is obtained by rotating point H by angle pi/3 about point A. Point E is the reflection of point C across line V. Let F be the midpoint of segment HL. Let K be the midpoint of segment CF. Line J passes through point K perpendicular to segment EP. Point W is the reflection of point P across line J. Find the distance between points H 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 OG = 5. Line Z passes through points O and G. Let P be the rotation of point O by angle pi/3 around point G. Line S is the perpendicular bisector of segment PO. Point N is the midpoint of segment PG. Let V be the line through point P parallel to line S. Lines Z and V intersect at point T. Find the measure of angle NPT, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SO = 6. Point W is the reflection of point O across point S. Line F is the angle bisector of angle SWO. Let C be the rotation of point W by angle pi/3 around point O. Let U be the midpoint of segment SC. Point R is the reflection of point U across line F. Let Z be the reflection of point U across point R. Point N is the reflection of point O across point Z. Let Q be the line through point Z perpendicular to line F. Point E is the reflection of point N across line Q. What is the length of segment EZ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RN = 4. Let Z be the midpoint of segment NR. Point F is obtained by rotating point N by angle pi/2 about point R. Let E be the rotation of point Z by angle 5pi/12 around point R. Let H be the midpoint of segment NZ. Point X is the reflection of point E across point Z. Line Q is the angle bisector of angle XNF. Point T is the midpoint of segment FR. Point D is obtained by rotating point Z by angle 5pi/12 about point H. Let L be the reflection of line Q through point D. Point K is the reflection of point E across line L. Let I be the angle bisector of angle RTD. Let Y be the midpoint of segment ZH. Point W is the reflection of point K across line I. Find the length of segment YW. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
CN = 10. Line I passes through point C perpendicular to segment CN. Point J is the reflection of point N across line I. Point Z is the midpoint of segment CN. Line E is the perpendicular bisector of segment CN. Let D be the rotation of point N by angle pi/6 around point Z. Point A is the reflection of point D across segment CN. Point Q is the reflection of point A across line E. Line B is the perpendicular bisector of segment JQ. Point R is the reflection of point C across line B. Find the length of segment RJ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let LG = 9. Point Z is the midpoint of segment GL. Let F be the midpoint of segment GZ. Point R is obtained by rotating point F by angle 11pi/12 about point Z. Let B be the vector from point R to point F. Let Q be the midpoint of segment GR. Let X be point Q translated by vector B. Let D be the rotation of point F by angle pi/12 around point Q. Let U be the midpoint of segment GX. Point O is the midpoint of segment LD. Line M passes through point U parallel to segment GL. Line W is the perpendicular bisector of segment GL. Let A be the perpendicular bisector of segment OU. Point N is the reflection of point Q across segment GL. Point J is the reflection of point L across line A. Line I is the reflection of line W across point J. Lines I and M intersect at point Y. What is the length of segment YN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OJ = 11. Let B be the midpoint of segment JO. Let V be the midpoint of segment JB. Point K is obtained by rotating point V by angle pi/12 about point J. Point S is the reflection of point V across point O. Let W be the rotation of point O by angle 7pi/12 around point B. Point N is the midpoint of segment SK. Point M is the reflection of point O across point W. Point C is the midpoint of segment NM. Line X passes through point B parallel to segment NM. Line X and segment KM intersect at point H. What is the length of segment CH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GR = 5. Line X is the perpendicular bisector of segment GR. Point A is on line X with |AG| = 7. Line W is the reflection of line X across point R. Let P be the line through point A parallel to segment GR. Point Q is the reflection of point R across line P. Point C is the reflection of point A across segment GR. Line K passes through points C and Q. Line K and segment GR intersect at point B. Let V be the reflection of point Q across point B. Line U passes through point V parallel to line W. Point M is the intersection of line U and segment GR. What is the distance between points Q 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 OE = 5. Let F be the rotation of point O by angle 7pi/12 around point E. Point C is the reflection of point F across segment EO. Line Z is the perpendicular bisector of segment FE. Point U is the midpoint of segment OF. Point P is the reflection of point U across segment EO. Let T be the vector from point C to point U. Let S be the line through point E perpendicular to segment EO. Point G is the reflection of point P across line S. Let J be point O translated by vector T. Point A is the reflection of point C across line S. Point N is obtained by rotating point U by angle 3pi/4 about point A. Point V is the reflection of point N across line Z. Point Y is the midpoint of segment VA. Point H is the reflection of point Y across point J. Point K is the reflection of point G across point H. Find the length of segment KH. 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 OM = 2. Let J be the rotation of point O by angle pi/6 around point M. Let Y be the rotation of point J by angle 3pi/4 around point O. Let H be the angle bisector of angle YMJ. Line V is the reflection of line H across point Y. Line P is the reflection of line V across point O. Let Q be the vector from point O to point J. Line X is the reflection of line H across line V. Let R be the reflection of point Y across point M. Line A passes through points J and M. Point B is the translation of point M by vector Q. Let D be the reflection of line A through point R. Point F is the reflection of point B across line P. Lines D and X intersect at point K. Let U be the midpoint of segment FJ. What is the length of segment KU? 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 JV = 11. Let S be the vector from point J to point V. Let U be point V translated by vector S. Let L be the rotation of point J by angle 11pi/12 around point U. Let P be the line through points U and L. Point Y is on segment VJ. Point I is the midpoint of segment YU. Point O is the reflection of point I across line P. Find the measure of angle IUO, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
YV = 10. Let W be the rotation of point Y by angle 3pi/4 around point V. Let X be the vector from point Y to point W. Point T is the translation of point W by vector X. Let J be the reflection of point W across point Y. Point A is the midpoint of segment VY. Let G be the line through point J perpendicular to segment TA. Line Q is the reflection of line G across point Y. Point B is the midpoint of segment AW. Point K is the reflection of point B across line Q. Point H is the intersection of segment TA and line Q. Find the distance between points H and K. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BO = 11. Let C be the rotation of point B by angle pi/3 around point O. Point D is obtained by rotating point C by angle pi/12 about point B. Point N is the midpoint of segment OC. Point L is the reflection of point N across point O. Point K is the reflection of point B across segment DL. Point P is the reflection of point O across segment DL. What is the distance between points K 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 BI = 5. Point A is the midpoint of segment BI. Let K be the line through point B perpendicular to segment BA. Point N is obtained by rotating point B by angle pi about point I. Point F is the reflection of point N across line K. Line R is the perpendicular bisector of segment BA. Let P be the perpendicular bisector of segment AN. Let Y be the reflection of line P across line R. Let M be the reflection of line P across line Y. Point L is the reflection of point N across line M. What is the distance between points L 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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UE = 12. Point V is obtained by rotating point U by angle 3pi/4 about point E. Point H is the midpoint of segment UE. Let X be the midpoint of segment HV. Let D be the reflection of point X across point H. Find the distance between points D 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 DE = 9. Point V is obtained by rotating point E by angle 11pi/12 about point D. Point S is the midpoint of segment VE. Line M is the perpendicular bisector of segment ED. Let Y be the rotation of point V by angle pi/4 around point D. Let Z be the reflection of line M through point V. Point X is the reflection of point E across line Z. Point H is the reflection of point S across segment ED. Let Q be the rotation of point X by angle pi/2 around point H. Point N is the midpoint of segment XQ. Let L be the midpoint of segment YQ. Let U be the reflection of point L across point S. Point W is the reflection of point U across segment ED. What is 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]
Let RO = 3. Let F be the rotation of point R by angle 5pi/12 around point O. Point D is the midpoint of segment FR. Let I be the rotation of point O by angle 5pi/6 around point R. Let N be the midpoint of segment DO. Let Q be the midpoint of segment IR. Let E be the reflection of point I across point N. Line Y is the angle bisector of angle QIE. Point H is the reflection of point N across line Y. Find the length of segment EH. 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 PE = 1. Let O be the vector from point P to point E. Line I is the perpendicular bisector of segment PE. Point U is the translation of point E by vector O. Let B be the rotation of point P by angle pi/12 around point E. Point J is the reflection of point U across line I. Let W be the vector from point B to point J. Point V is on line I. Let N be point E translated by vector W. Let R be the vector from point N to point J. Let Z be point J translated by vector R. Let D be the line through point V parallel to segment PE. Line S is the angle bisector of angle UNZ. Let M be the line through point B perpendicular to line D. Point L is the reflection of point Z across line S. Point T is the reflection of point E across line M. What is the length of segment LT? 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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KP = 12. Let R be the perpendicular bisector of segment PK. Point B is on line R with |BP| = 12. Point D is the reflection of point K across point P. Let O be the midpoint of segment PB. Let E be the midpoint of segment OD. Point W is the reflection of point E across point P. What is the length of segment OW? 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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IW = 9. Point Y is the reflection of point W across point I. Point G is obtained by rotating point W by angle 2pi/3 about point I. Let C be the midpoint of segment GI. Let N be the rotation of point I by angle pi/12 around point G. Let P be the line through point W perpendicular to segment GY. Point T is the reflection of point Y across line P. Point K is the midpoint of segment TC. Find the distance between points K and N. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HX = 5. Let J be the line through points H and X. Let Z be the rotation of point X by angle pi/6 around point H. Let I be the reflection of point X across point H. Point E is the reflection of point I across point Z. Line N is the perpendicular bisector of segment HI. Let T be the midpoint of segment EH. Line S passes through point T parallel to line N. Point Q is the intersection of lines J and S. Find the distance between points H and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EK = 9. Let X be the reflection of point K across point E. Let Y be the rotation of point K by angle pi/4 around point X. Let P be the reflection of point Y across point K. Let G be the reflection of point K across point Y. Point A is the midpoint of segment KE. Point C is the reflection of point A across point G. Point N is the reflection of point C across segment KE. What is the distance between points N 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 RA = 4. Point Z is obtained by rotating point R by angle 5pi/12 about point A. Line G passes through point Z perpendicular to segment RA. Line L passes through point A perpendicular to segment AZ. Let F be the reflection of point R across point A. Line D is the reflection of line G across point F. Let Y be the midpoint of segment RA. Point U is the reflection of point Y across line L. Point K is the reflection of point Z across line D. Point W is the reflection of point U across segment AZ. What is the length of segment KW? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
DV = 2. Point Z is obtained by rotating point D by angle 7pi/12 about point V. Point Q is the reflection of point V across point D. Let R be the reflection of point Z across point Q. Let X be the reflection of point Q across point V. Let B be the perpendicular bisector of segment RV. Point G is the reflection of point Q across line B. Find the length of segment GX. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let SG = 3. Let J be the midpoint of segment GS. Point V is obtained by rotating point S by angle 3pi/4 about point J. Let F be the angle bisector of angle GVS. Let I be the rotation of point S by angle 11pi/12 around point V. Point Y is the reflection of point I across segment VG. Line C is the reflection of line F across point I. Point O is the reflection of point I across line C. Point R is the reflection of point I across segment YV. Find the length of segment RO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VL = 6. Let P be the reflection of point L across point V. Point R is the midpoint of segment LV. Line M is the angle bisector of angle RVP. Let Z be the reflection of point R across point P. Point J is the reflection of point Z across line M. Let N be the rotation of point P by angle pi/6 around point J. Let G be the midpoint of segment ZN. Let D be the reflection of line M through point J. Point B is the midpoint of segment LR. Point T is the reflection of point G across line D. Find 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}


[User]
HD = 3. Point G is the reflection of point H across point D. Let S be the midpoint of segment GD. Let O be the perpendicular bisector of segment DH. Point C is the reflection of point G across line O. Point P is the midpoint of segment GS. What is the distance between points C and P? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let FP = 2. Let I be the vector from point P to point F. Let G be the rotation of point F by angle pi around point P. Let E be the perpendicular bisector of segment GF. Let Z be the reflection of point F across point G. Let H be the perpendicular bisector of segment GP. Point M is the reflection of point Z across line H. Line W is the angle bisector of angle ZFM. Let K be the reflection of line E across line H. Point T lies on line E at distance 5 from point Z. Point C is the reflection of point Z across line W. Point J is the translation of point T by vector I. Let S be the reflection of point M across point T. Point Q is the reflection of point S across line K. Point V is the reflection of point P across segment JC. Let D be the line through point T perpendicular to segment VQ. Line U passes through point G perpendicular to line D. Point N is the reflection of point S across line U. What is the length of segment ZN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NM = 10. Point T is the midpoint of segment MN. Point D is obtained by rotating point N by angle 5pi/6 about point M. Point F is the midpoint of segment DN. Point X is the reflection of point D across segment FT. Line K passes through points T and M. Let V be the line through point X perpendicular to line K. Point W is the reflection of point M across line V. What is the length of segment WX? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let SN = 1. Let K be the rotation of point S by angle 5pi/12 around point N. Let C be the midpoint of segment SN. Let B be the rotation of point C by angle 5pi/6 around point N. Point M is the midpoint of segment CK. Let T be the rotation of point B by angle 7pi/12 around point C. Let R be the reflection of point T across point M. Point U is obtained by rotating point S by angle pi/3 about point N. Let D be the reflection of point M across point S. Point X is the reflection of point R across segment KD. Point V is the intersection of segments TU and DN. Find the distance between points X and V. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QA = 4. Point K is obtained by rotating point A by angle 2pi/3 about point Q. Let T be the perpendicular bisector of segment KQ. Line X passes through point Q parallel to line T. Let Z be the rotation of point Q by angle 5pi/12 around point K. Point H is the midpoint of segment KA. Let I be the line through point H parallel to line X. Let E be the line through point Z parallel to segment KA. Lines E and I intersect at point M. What is the distance between points M and K? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JV = 4. Let U be the reflection of point J across point V. Let Z be the rotation of point U by angle 3pi/4 around point J. Line G passes through points J and Z. Let F be the perpendicular bisector of segment JV. Let R be the reflection of line G across line F. Lines G and R intersect at point O. Point P is obtained by rotating point V by angle 5pi/6 about point J. Line E is the reflection of line F across point J. Line D passes through point Z parallel to line E. Line Q is the perpendicular bisector of segment OP. Line I passes through point V parallel to line Q. Line T is the reflection of line R across line I. Lines T and D intersect at point W. Line I and segment OW intersect at point K. Find the distance between points K and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WK = 6. Line B passes through points K and W. Point I is obtained by rotating point K by angle 7pi/12 about point W. Point Y is the midpoint of segment WK. Let J be the perpendicular bisector of segment YK. Let C be the line through points I and K. Line H passes through point K parallel to line J. Point G is obtained by rotating point W by angle pi/2 about point K. Let P be the midpoint of segment IY. Let S be the midpoint of segment GI. Point R is the reflection of point Y across line H. Let V be the line through point W parallel to line C. Let F be the line through points K and P. Line E is the angle bisector of angle SRK. Let N be the reflection of line B across line E. Line T is the reflection of line N across line V. Lines F and T intersect at point Q. What is the distance between points Q 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 NW = 7. Let V be the rotation of point N by angle pi/6 around point W. Line G is the angle bisector of angle VWN. Let U be the angle bisector of angle WVN. Let O be the reflection of point V across point W. Line I is the reflection of line U across line G. Point Q is the reflection of point W across line I. Let T be the reflection of point N across point W. Point L is the reflection of point O across segment VN. Point M is the reflection of point V across segment TN. Let B be the midpoint of segment ML. Find the length of segment BQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
TG = 12. Let V be the midpoint of segment GT. Let E be the rotation of point V by angle 7pi/12 around point T. Let O be the reflection of point T across point G. Point M is obtained by rotating point V by angle pi/3 about point T. Point Q is obtained by rotating point G by angle 2pi/3 about point E. Let R be the rotation of point Q by angle 7pi/12 around point V. Let I be the midpoint of segment VO. Point Y is the midpoint of segment EQ. Point H is the reflection of point I across point Y. Point F is the reflection of point H across segment GV. Point N is the reflection of point M across segment OR. Point X is the midpoint of segment NQ. 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]
LO = 7. Let F be the line through points O and L. Point M is obtained by rotating point L by angle pi about point O. Line R passes through point M perpendicular to line F. Let S be the angle bisector of angle LOM. Let P be the reflection of point L across point M. Line K passes through point P parallel to line S. Point A is the reflection of point M across point P. Point H is the midpoint of segment PA. Let Y be the rotation of point L by angle 5pi/6 around point H. Point T is the reflection of point H across line R. Point W is the reflection of point T across line K. Let V be the midpoint of segment YW. Let X be the reflection of point M across point V. What is 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 OW = 1. Point V is the midpoint of segment WO. Point P is obtained by rotating point W by angle pi about point O. Line I is the perpendicular bisector of segment WP. Line A is the reflection of line I across point V. Line Y is the reflection of line A across line I. Point U is on line A. Line M is the reflection of line Y across line A. Point R is the reflection of point U across line M. What is the length of segment UR? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AC = 1. Let W be the rotation of point A by angle pi/6 around point C. Let V be the reflection of point W across point C. Point Z is the midpoint of segment AW. Line K passes through point W parallel to segment AC. Point D is the midpoint of segment VC. Let X be the reflection of point V across point W. Point F is the reflection of point Z across point X. Point Y is the reflection of point F across point A. Point U is the midpoint of segment YD. Point G is the intersection of line K and segment UF. What is the distance between points G and U? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XF = 7. Let D be the reflection of point F across point X. Let Z be the perpendicular bisector of segment XD. Line P passes through point D parallel to line Z. Let K be the line through point F perpendicular to segment XD. Point C lies on line K at distance 10 from point X. Point M lies on line P at distance 7 from point X. Point V is on segment XD. Line L is the perpendicular bisector of segment FC. Point J is on line L. Line N is the perpendicular bisector of segment MV. Let W be the reflection of point V across point J. Point H is the reflection of point F across line N. Let G be the midpoint of segment WH. What is the distance between points J 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 JZ = 6. Point R is the midpoint of segment JZ. Line F is the perpendicular bisector of segment ZR. Point Y is the midpoint of segment RZ. Point O is obtained by rotating point Z by angle pi/6 about point R. Let D be the midpoint of segment JR. Let L be the rotation of point O by angle 7pi/12 around point Z. Let M be the reflection of point Y across point L. Let S be the reflection of line F through point D. Line C passes through point M parallel to segment RZ. Lines S and C intersect at point Q. What is 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]
Let JH = 12. Point R is the reflection of point H across point J. Point K is the midpoint of segment RJ. Line A is the perpendicular bisector of segment KJ. Line D passes through point K parallel to line A. Point Q is the reflection of point H across line D. Point Z is the midpoint of segment QK. Find the length of segment KZ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VZ = 6. Let R be the reflection of point V across point Z. Point K is obtained by rotating point Z by angle pi about point R. Let W be the midpoint of segment ZR. Let Q be the midpoint of segment WK. Let F be the perpendicular bisector of segment QV. Point X is the reflection of point Z across line F. 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]
Let KC = 2. Line H passes through points K and C. Let G be the reflection of point C across point K. Let E be the angle bisector of angle CKG. Line X is the reflection of line E across point C. Let F be the reflection of point C across point G. Point L lies on line X at distance 5 from point C. Line J is the angle bisector of angle FGL. Let W be the reflection of line H across line J. Point R is the reflection of point C across line W. Point P is the reflection of point R across point F. Find 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]
Let YO = 1. Line J passes through points Y and O. Let H be the reflection of point Y across point O. Let F be the rotation of point Y by angle pi/3 around point H. Point D is obtained by rotating point F by angle pi/3 about point H. Point W is the midpoint of segment DY. Point B is the reflection of point O across point F. Let G be the rotation of point Y by angle 5pi/6 around point W. Let C be the perpendicular bisector of segment GW. Let E be the reflection of line C across line J. Point U is the reflection of point H across line E. Let I be the midpoint of segment FB. Let R be the reflection of point U across point I. What is the length of segment RU? 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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YN = 10. Point Z is the reflection of point N across point Y. Let J be the perpendicular bisector of segment ZY. Let L be the reflection of line J through point Y. Point V is on line L with |VN| = 5. Line S is the perpendicular bisector of segment VZ. Point F is the reflection of point Y across line S. Point P is the reflection of point V across segment YF. Let G be the midpoint of segment YF. Let B be the reflection of point P across point Z. Let T be the midpoint of segment BG. Point E is the reflection of point T across segment YF. Point M is the midpoint of segment EG. What is the distance between points M and B? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
MT = 1. Point X is the reflection of point T across point M. Let G be the rotation of point X by angle pi/2 around point M. Let L be the midpoint of segment TG. Let J be the reflection of point G across point T. Let C be the midpoint of segment XL. Point F is the reflection of point J across point C. What is the distance between points G 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 WE = 10. Line Z is the perpendicular bisector of segment EW. Let L be the midpoint of segment EW. Let A be the rotation of point W by angle 5pi/6 around point E. Point F is obtained by rotating point W by angle pi/2 about point L. Line C passes through point A parallel to line Z. Point H is the reflection of point W across line C. Point T is the reflection of point A across point L. Let Y be the reflection of point T across point F. Find the distance between points H and Y. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let HS = 7. Let HD = 10. Let V be the rotation of point H by angle 5pi/6 around point S. Point W lies on segment SH. Let J be the perpendicular bisector of segment SH. Point Q is the reflection of point H across segment DW. Line B passes through point Q perpendicular to segment SH. Point Z is the reflection of point V across line J. Let P be the line through point Z perpendicular to line B. Point M is the reflection of point W across line P. Find the distance between points W and M. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LG = 8. Let W be the reflection of point L across point G. Let E be the perpendicular bisector of segment LG. Point Q is obtained by rotating point L by angle pi/2 about point G. Point H is the reflection of point Q across line E. Point Y is the midpoint of segment WG. Point Z is the midpoint of segment LH. Let N be the rotation of point Y by angle pi/12 around point H. Let O be the midpoint of segment QZ. Point S is the reflection of point N across segment LH. Point U is the reflection of point O across segment LH. Let R be the midpoint of segment SQ. Find the length of segment UR. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RY = 3. Let K be the perpendicular bisector of segment YR. Point C is on segment YR. Let X be the angle bisector of angle RCY. Point I lies on line X. Let N be the rotation of point Y by angle pi around point C. Let Q be the line through point I perpendicular to line K. Point D is the midpoint of segment YR. Point L is the reflection of point R across line Q. Point G is the reflection of point N across point I. Point S is the reflection of point G across segment LD. Point J is the reflection of point S across segment CL. Find 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]
UM = 5. Let T be the perpendicular bisector of segment UM. Let Q be the line through point M perpendicular to segment UM. Point G is the reflection of point M across point U. Line W is the reflection of line Q across line T. Let S be the reflection of line W across line Q. Point V lies on line S at distance 8 from point G. Point O is the reflection of point U across line Q. Line L is the reflection of line Q across line W. Point X is the reflection of point V across line W. Line Y passes through point U parallel to segment OX. Point C is the intersection of lines Y and L. Find the length of segment CO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GI = 4. Point F is the midpoint of segment IG. Point D is obtained by rotating point I by angle 3pi/4 about point G. Line P passes through point I perpendicular to segment DG. Point C is the reflection of point G across line P. Line X passes through point D parallel to segment GF. Point O is the reflection of point C across line X. Let H be the angle bisector of angle FDC. Point A is the reflection of point O across segment DG. Point S is the reflection of point A across line H. Find the distance between points S 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 ZQ = 12. Point K is obtained by rotating point Z by angle 2pi/3 about point Q. Let J be the reflection of point Q across point K. Point N is obtained by rotating point J by angle pi/2 about point Q. Let V be the midpoint of segment JN. Point F is the reflection of point V across segment JZ. Point O is the midpoint of segment ZV. Point E is the reflection of point O across segment JZ. Point M is the reflection of point E across segment JF. What is the distance between points F 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 BJ = 7. Let V be the line through points B and J. Point D is obtained by rotating point J by angle 11pi/12 about point B. Point Y is the reflection of point J across point B. Let S be the perpendicular bisector of segment DB. Let R be the reflection of line S through point J. Let T be the midpoint of segment YD. Point P is the intersection of lines V and R. Point K is the reflection of point T across line V. Line H is the perpendicular bisector of segment PK. Point E is the reflection of point T across line H. Let C be the midpoint of segment PE. Point N is the reflection of point C across line H. Find the length of segment KN. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HV = 1. Let C be the rotation of point V by angle 5pi/6 around point H. Let F be the line through points C and H. Line M passes through point V parallel to line F. Let K be the midpoint of segment HV. Point P is the reflection of point K across point C. Let B be the rotation of point C by angle 5pi/6 around point V. Point S is the reflection of point P across line M. Line D is the angle bisector of angle VBC. Line Q passes through points S and P. Point R is the reflection of point H across point S. Point W is the reflection of point V across point S. Line A is the reflection of line Q across line D. Let O be the line through point W parallel to line D. Let L be the reflection of line O through point P. Point Y is the intersection of lines L and A. Point I is the reflection of point K across segment WR. Point J is the reflection of point Y across segment WR. Find the length of segment IJ. 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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KF = 1. Point M is obtained by rotating point F by angle 2pi/3 about point K. Point R is the midpoint of segment FK. Let G be the reflection of point K across point F. Let Q be the reflection of point K across point M. Point J is the midpoint of segment FQ. Point X is the midpoint of segment RQ. Let N be the rotation of point G by angle 11pi/12 around point K. Line O is the perpendicular bisector of segment FK. Point Z is the reflection of point J across line O. Point S is the reflection of point Z across segment FQ. Let L be the midpoint of segment XN. What is the length of segment SL? Express your answer as a decimal number rounded to 2 decimal places.

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


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WX = 8. Point Y is obtained by rotating point W by angle pi/3 about point X. Point Q is the reflection of point Y across point W. Point P is the midpoint of segment YX. Point S is the reflection of point Y across segment XW. Let T be the reflection of point Q across point P. Let B be the line through points T and S. Point O is the reflection of point P across line B. Point I is the reflection of point W across line B. What is the length of segment IO? 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 QF = 4. Let V be the rotation of point Q by angle 11pi/12 around point F. Let J be the vector from point Q to point V. Point P is the translation of point F by vector J. Let T be the reflection of point P across point V. Let O be the rotation of point F by angle pi/12 around point T. Let D be the rotation of point Q by angle 5pi/6 around point P. Let E be point O translated by vector J. Let B be the perpendicular bisector of segment EQ. Point K is the reflection of point T across line B. Let W be the midpoint of segment OK. Let Z be the midpoint of segment WD. What is the distance between points K and Z? Express your answer as a decimal number rounded to 2 decimal places.

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


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VT = 2. Point C is the reflection of point T across point V. Let N be the angle bisector of angle TVC. Let Z be the rotation of point V by angle 7pi/12 around point C. Let F be the reflection of point V across point C. Let I be the vector from point F to point T. Let H be the line through points Z and F. Point X is the translation of point Z by vector I. Line S passes through point X perpendicular to line N. Line U passes through points X and F. Point P is the reflection of point V across line S. Let J be the reflection of line H across line S. Point E is the intersection of lines J and U. What is the distance between points P 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 TG = 8. Let N be the perpendicular bisector of segment TG. Line L is the reflection of line N across point T. Point V lies on line L at distance 4 from point T. Point B is the midpoint of segment GT. Let J be the midpoint of segment VB. Point W is the reflection of point J across segment GT. Let M be the reflection of point W across point G. Find the length of segment WM. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ZX = 12. Let H be the midpoint of segment XZ. Point J is obtained by rotating point Z by angle pi/6 about point X. Let L be the perpendicular bisector of segment JH. Point T is the reflection of point Z across line L. Let B be the midpoint of segment TJ. Let F be the midpoint of segment ZJ. Point E is the reflection of point F across point T. What is the distance between points E and B? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OI = 7. Point G is obtained by rotating point I by angle pi/12 about point O. Let N be the midpoint of segment IO. Line H passes through point I perpendicular to segment OG. Line D is the reflection of line H across point G. Point R is the reflection of point N across line D. Let L be the midpoint of segment OR. Find the distance between points G and L. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GU = 12. Line X passes through point G perpendicular to segment UG. Line M passes through point U parallel to line X. Let O be the midpoint of segment GU. Let Y be the reflection of line M across line X. Point N is the reflection of point G across line Y. Point B is the midpoint of segment NO. What is the distance between points B and N? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BS = 11. Point X is obtained by rotating point B by angle 5pi/6 about point S. Line R passes through points X and B. Point L is the midpoint of segment SX. Point I is the reflection of point L across line R. What is the distance between points I 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 IU = 6. Let O be the rotation of point U by angle 5pi/6 around point I. Let X be the reflection of point O across point U. Point T is on segment UI. Line D is the perpendicular bisector of segment TX. Point E is the reflection of point I across line D. Find the distance between points E 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 WT = 2. Point H is obtained by rotating point T by angle pi/3 about point W. Point B is obtained by rotating point W by angle 5pi/12 about point H. Let Z be the reflection of point T across point B. Point P is the reflection of point H across point Z. Point S is the reflection of point P across segment BT. Find the distance between points H and S. Express your answer as a decimal number rounded to 2 decimal places.

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


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VC = 4. Point N is the reflection of point V across point C. Line Y is the angle bisector of angle CVN. Let W be the angle bisector of angle NCV. Let S be the rotation of point V by angle 5pi/6 around point C. Point H is the midpoint of segment SC. Let Z be the midpoint of segment HN. Point B is the reflection of point Z across line Y. Point U is the reflection of point N across segment CB. Line E is the perpendicular bisector of segment CB. Line J is the reflection of line E across point H. Let I be the line through point C parallel to line J. Let R be the reflection of line I across line W. Point K is the reflection of point U across line R. What is the length of segment KZ? Express your answer as a decimal number rounded to 2 decimal places.

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


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Let JM = 8. Let X be the rotation of point J by angle pi/2 around point M. Let P be the midpoint of segment XM. Let O be the reflection of point J across point X. Point D is the reflection of point O across segment XM. Let Z be the midpoint of segment PJ. Point A is the midpoint of segment XD. 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}


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VB = 3. Let L be the reflection of point V across point B. Let M be the vector from point V to point B. Point Y is the reflection of point L across point V. Point D is the translation of point L by vector M. Line Q is the angle bisector of angle LBY. Point T is the reflection of point Y across line Q. Point S is on line Q with |SB| = 12. Line I is the angle bisector of angle SDB. Point O is the reflection of point L across point S. Point P is the reflection of point V across line I. Point K is the reflection of point O across point T. Point U is the midpoint of segment KP. 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}


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IK = 7. Let C be the vector from point K to point I. Let O be the rotation of point K by angle 11pi/12 around point I. Let F be the rotation of point I by angle 7pi/12 around point O. Let R be the reflection of point F across point K. Line L is the angle bisector of angle OKR. Point N is the translation of point O by vector C. Let Q be the line through points R and K. Let Y be the angle bisector of angle ORF. Let X be the reflection of point O across point N. Let E be the reflection of line Y across line L. Lines E and Q intersect at point A. What is the distance between points X and A? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QP = 10. Let M be the vector from point Q to point P. Point N is the translation of point P by vector M. Let A be the line through points P and N. Point O is obtained by rotating point Q by angle pi/12 about point P. Point W is the midpoint of segment OQ. Point V is the reflection of point P across point O. Line U is the reflection of line A across point V. Let D be the line through points O and N. Point R is the midpoint of segment QW. Point Y is the reflection of point W across line D. Point Z is the reflection of point Y across line U. Let B be the midpoint of segment ZR. What is the length of segment BZ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OE = 6. Point K is the midpoint of segment EO. Point Q is the reflection of point O across point E. Point X is the midpoint of segment KO. Let R be the rotation of point E by angle 5pi/6 around point X. Let D be the rotation of point O by angle 5pi/6 around point Q. Let P be the reflection of point O across point Q. Let J be the midpoint of segment PR. Line L passes through point X perpendicular to segment ED. Let V be the line through point Q parallel to line L. Point H is the reflection of point J across line V. What is the distance between points J 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 HT = 10. Let M be the perpendicular bisector of segment TH. Point X is on line M with |XH| = 8. Point P is the midpoint of segment XT. Line R passes through point T perpendicular to segment HX. Let Y be the reflection of point P across point X. Point U is the reflection of point Y across line R. What is 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]
WT = 11. Let R be the midpoint of segment WT. Point S is the midpoint of segment RT. Point Z is the midpoint of segment RS. Find the distance between points Z and R. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VY = 5. Let Q be the vector from point Y to point V. Let L be the midpoint of segment VY. Line N is the perpendicular bisector of segment VY. Point J lies on line N at distance 3 from point Y. Point A is the translation of point J by vector Q. Point S is the translation of point L by vector Q. Point C is the midpoint of segment LA. Let W be the line through point C parallel to segment AJ. Point M is on segment AJ. Let I be the reflection of line W through point M. Point P is the reflection of point A across line I. Find the distance between points S and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let EV = 7. Let P be the vector from point V to point E. Let Z be the reflection of point V across point E. Point H is obtained by rotating point V by angle pi/2 about point E. Let L be the rotation of point Z by angle 11pi/12 around point E. Let I be point Z translated by vector P. Line Y is the perpendicular bisector of segment ZH. Line X is the reflection of line Y across point H. Let O be the rotation of point I by angle pi around point E. Let T be the reflection of point L across point I. Let R be the reflection of point T across point O. Let S be the rotation of point R by angle pi/6 around point T. Let M be the reflection of line X through point S. Point N is the reflection of point O across line M. Point Q is the reflection of point T across segment NI. Find the length of segment QL. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ZA = 1. Let F be the perpendicular bisector of segment ZA. Let O be the midpoint of segment ZA. Point K is on line F with |KO| = 2. Point R is the midpoint of segment ZO. Point W is the midpoint of segment RA. Line G is the perpendicular bisector of segment KA. Let D be the line through point O parallel to line G. Point X is the intersection of line D and segment KA. Find the length of segment XW. 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 = 4. Point B is the midpoint of segment TP. Let V be the reflection of point B across point P. Line M is the perpendicular bisector of segment TB. Point E is obtained by rotating point V by angle pi/2 about point T. Point U is obtained by rotating point E by angle 11pi/12 about point P. Let A be the line through point E perpendicular to segment UT. Point N is the reflection of point U across line A. Point Y is the reflection of point N across line M. Point W is the reflection of point Y across segment UT. What is the distance between points Y 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 CO = 7. Let J be the rotation of point O by angle pi/2 around point C. Let U be the midpoint of segment CJ. Let Q be the rotation of point U by angle pi/2 around point O. Point H is obtained by rotating point Q by angle pi/4 about point O. Let D be the reflection of point U across point O. Point V is the reflection of point H across point D. Let B be the midpoint of segment CD. What is the length of segment VB? 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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HR = 3. Point G is the reflection of point R across point H. Line C passes through points R and G. Let P be the rotation of point H by angle 3pi/4 around point R. Let S be the rotation of point R by angle pi/3 around point H. Let N be the reflection of point G across point P. Let D be the line through point S parallel to line C. Let E be the midpoint of segment HN. Point W is the reflection of point P across line D. Let K be the midpoint of segment WE. Point I is the midpoint of segment KP. What is the distance between points I 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 BX = 5. Point Z is obtained by rotating point B by angle pi/12 about point X. Let D be the reflection of point B across point Z. Line Q is the perpendicular bisector of segment BX. Line O passes through point D parallel to line Q. Let V be the line through point B perpendicular to segment DX. Let K be the reflection of line V through point D. Point T is the intersection of lines K and O. Let F be the midpoint of segment DX. Point W is the midpoint of segment TF. Find the length of segment WZ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VG = 3. Point E is obtained by rotating point G by angle 11pi/12 about point V. Let W be the midpoint of segment GV. Line Z is the angle bisector of angle VWE. Let S be the midpoint of segment EV. Point R is the reflection of point G across segment VE. Point P is the reflection of point S across line Z. Let N be the midpoint of segment PR. What is the distance between points R and N? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XO = 9. Point Y is the reflection of point O across point X. Point M is the midpoint of segment XY. Line S is the angle bisector of angle YMX. Let N be the rotation of point O by angle pi/4 around point M. Let U be the line through points Y and M. Line W is the reflection of line S across point N. Let Z be the reflection of point X across point Y. Line L is the angle bisector of angle NMX. Point B is the midpoint of segment YN. Point P is the intersection of lines U and W. Point T is the midpoint of segment PN. Let G be the reflection of line L through point N. Point R is the reflection of point T across line G. Point K is the reflection of point Z across point R. What is the length of segment BK? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let SJ = 12. Point U is obtained by rotating point S by angle pi/2 about point J. Point W is the reflection of point S across point J. Let A be the perpendicular bisector of segment UJ. Point C is the midpoint of segment JW. Let Y be the midpoint of segment UJ. Let V be the reflection of point U across point W. Line B passes through point V perpendicular to line A. Point N is the reflection of point J across line B. Point K is the reflection of point Y across point N. Find the length of segment CK. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let QR = 11. Let X be the reflection of point R across point Q. Let C be the midpoint of segment QR. Line I is the perpendicular bisector of segment QC. Let O be the rotation of point R by angle 7pi/12 around point Q. Let A be the reflection of line I through point Q. Let H be the reflection of point Q across point R. Point Z is the midpoint of segment XQ. Let S be the midpoint of segment XO. Point D is the reflection of point Z across point S. Point F is the reflection of point D across line A. Point M is the reflection of point F across segment XR. Find the length of segment HM. 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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JB = 10. Point H is the midpoint of segment BJ. Point N is the midpoint of segment JH. Point S is obtained by rotating point N by angle pi/4 about point B. Let L be the perpendicular bisector of segment HS. Point I is the midpoint of segment BH. Point Z is the reflection of point N across line L. What is the length of segment ZI? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OI = 10. Line K is the perpendicular bisector of segment OI. Line G passes through point I parallel to line K. Let C be the midpoint of segment OI. Line V is the reflection of line K across line G. Let H be the line through points C and I. Point U is obtained by rotating point O by angle 3pi/4 about point I. Point M is obtained by rotating point C by angle pi/4 about point I. Let S be the reflection of point C across point I. Let F be the perpendicular bisector of segment IU. Let R be the reflection of line F across line V. Point L is the intersection of lines R and H. Point X is the reflection of point C across point S. Lines R and G intersect at point J. Point Z is the reflection of point L across point U. Point D is the midpoint of segment MJ. Point E is the midpoint of segment DX. What is the length of segment EZ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QT = 10. Let Y be the reflection of point T across point Q. Point F is the midpoint of segment YQ. Point D is the midpoint of segment QT. Let E be the rotation of point Q by angle pi/6 around point F. Let M be the perpendicular bisector of segment YE. Line U is the reflection of line M across point Y. Let G be the rotation of point E by angle pi/12 around point D. Point S is the reflection of point T across point Y. Let A be the midpoint of segment SY. Let V be the reflection of line M across line U. Point B is the reflection of point G across point E. Point L is the midpoint of segment FB. Let I be the line through point B parallel to line M. Let P be the angle bisector of angle BYA. Let R be the line through point L perpendicular to line I. Let Z be the rotation of point A by angle pi/3 around point B. Line W is the reflection of line R across line P. Lines W and V intersect at point O. Line X is the reflection of line I across point S. Let H be the midpoint of segment ZL. Point C is the reflection of point O across line X. Point K is the midpoint of segment CH. What is the distance between points K 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 IJ = 12. Point E is the midpoint of segment IJ. Let A be the rotation of point J by angle 7pi/12 around point E. Z is placed 12 units away from J. Let H be the midpoint of segment AI. Point B is the midpoint of segment ZE. Point K is the reflection of point B across point A. Let D be the perpendicular bisector of segment BK. Point L is the reflection of point H across line D. 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]
SD = 7. Let X be the reflection of point D across point S. Point Z is the reflection of point X across point D. Let G be the reflection of point D across point X. Let V be the midpoint of segment ZD. Line M passes through point G perpendicular to segment XG. Point P is the midpoint of segment VG. Point K is the midpoint of segment DP. Point H is the reflection of point X across line M. Point O is the reflection of point H across point Z. Let I be the reflection of point K across point D. What is the distance between points O and I? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WL = 1. Let D be the line through points L and W. Point S is the midpoint of segment WL. Point U is obtained by rotating point W by angle 7pi/12 about point S. Let H be the reflection of line D through point U. Let O be the reflection of line H across line D. Point M is on line O with |MS| = 12. Line Z is the perpendicular bisector of segment SU. Let C be the reflection of line Z through point S. Line X passes through point M parallel to line C. Let Y be the line through point S perpendicular to line X. Point R is the reflection of point L across line Y. Find the distance between points R and W. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let SC = 9. Line P is the perpendicular bisector of segment CS. Let Y be the rotation of point C by angle pi/4 around point S. Let D be the rotation of point C by angle 7pi/12 around point Y. Point E is the reflection of point D across line P. Line U passes through points D and S. Let G be the reflection of point Y across point E. Point X is the midpoint of segment EC. Line I is the reflection of line U across point Y. Point K is the reflection of point X across line I. Let A be the line through points D and Y. Let T be the line through point K perpendicular to line A. Point W is the midpoint of segment GX. Point V is the reflection of point W across line T. Let M be the reflection of point V across point X. What is 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]
VN = 1. Line U is the perpendicular bisector of segment NV. Line Y is the reflection of line U across point V. Line L passes through point N parallel to line U. Point R is the reflection of point V across line L. Point X is obtained by rotating point V by angle 3pi/4 about point N. Let D be the line through points N and X. Line H is the reflection of line U across line D. Lines H and Y intersect at point O. Let E be the midpoint of segment OR. Point J is the midpoint of segment EN. Point P is the midpoint of segment EJ. What is 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 RZ = 2. Point L is obtained by rotating point R by angle pi/2 about point Z. Point X is the reflection of point L across point Z. Point N is obtained by rotating point Z by angle 7pi/12 about point R. Let S be the rotation of point X by angle pi/6 around point L. Point E is the reflection of point L across point S. Line K passes through point R parallel to segment XN. Point J is the reflection of point E across line K. Let G be the reflection of point X across point J. What is the distance between points G and S? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
MF = 10. Point X is the midpoint of segment FM. Let B be the rotation of point M by angle 3pi/4 around point X. Let E be the reflection of point M across point B. Point R is the midpoint of segment XB. Point G is the reflection of point B across point F. Point P is the reflection of point G across point R. Find the distance between points E 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 GP = 2. Let O be the perpendicular bisector of segment PG. Point K is the midpoint of segment GP. Line U passes through point P parallel to line O. Point Z is on line U with |ZK| = 9. Line S is the angle bisector of angle PGZ. Line W is the reflection of line U across line S. Point N is the reflection of point Z across line W. Let E be the reflection of point G across point Z. Point I is the reflection of point E across segment NP. Find the length of segment NI. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OX = 7. Let Y be the midpoint of segment OX. Let B be the rotation of point O by angle 7pi/12 around point Y. Point R is the reflection of point B across point X. Let V be the reflection of point B across point Y. Line C is the angle bisector of angle VRX. Point W is the reflection of point O across point V. Point M is the reflection of point Y across line C. Point H is the reflection of point V across point M. Let F be the midpoint of segment WV. What is the length of segment FH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JR = 4. Point S is the midpoint of segment JR. Point D is obtained by rotating point R by angle pi/12 about point S. Point E is the midpoint of segment DR. Line F is the angle bisector of angle SEJ. Let K be the vector from point J to point E. Let G be point D translated by vector K. Point N is obtained by rotating point R by angle 5pi/6 about point G. Let W be the midpoint of segment RN. Point Z is the reflection of point W across line F. Point B is the reflection of point W across point N. Point H is the midpoint of segment BN. Point Y is the midpoint of segment BZ. What is the distance between points H and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let AK = 5. Let W be the vector from point A to point K. Let J be point K translated by vector W. Point F is the midpoint of segment JK. Point M is the translation of point J by vector W. Line I passes through point K perpendicular to segment AK. Let O be the reflection of line I through point F. Line P is the perpendicular bisector of segment AK. Point D is on line I with |DM| = 6. Let G be the reflection of line O across line P. Line U passes through point J parallel to segment MD. Point E is the intersection of lines U and G. What is 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 NE = 1. Let A be the perpendicular bisector of segment NE. Point W is the reflection of point E across point N. Point P is obtained by rotating point E by angle 2pi/3 about point W. Point B is obtained by rotating point W by angle 11pi/12 about point N. Let O be the midpoint of segment PN. Point K is the midpoint of segment WO. Point Y is the midpoint of segment PB. Point D is the reflection of point K across line A. Let Z be the reflection of point D across point P. What is the length of segment YZ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YS = 1. Let N be the midpoint of segment YS. Let O be the rotation of point S by angle 5pi/6 around point Y. Line U passes through points O and N. Point P is obtained by rotating point Y by angle pi/4 about point O. Let W be the rotation of point P by angle 5pi/6 around point S. Let R be the midpoint of segment OY. Point L is the midpoint of segment WO. Point C is the reflection of point R across line U. Find the distance between points L and C. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let SR = 9. Let E be the vector from point R to point S. Let SI = 9. Let D be the line through points I and R. Let U be the midpoint of segment IS. Point K is obtained by rotating point U by angle 2pi/3 about point S. Let Z be point S translated by vector E. Point B is the reflection of point Z across line D. Point N is the reflection of point I across segment BU. Let H be the midpoint of segment IK. Point O is the reflection of point H across segment BU. What is the distance between points N and O? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let TI = 6. Let E be the line through points I and T. Let A be the line through point T perpendicular to line E. Line R passes through point I parallel to line A. Let X be the perpendicular bisector of segment TI. Point L is on line R with |LT| = 11. Point D is the reflection of point L across line X. Point Z is the midpoint of segment LD. Point W is the midpoint of segment ZI. Let V be the reflection of point W across point Z. What is the distance between points V and T? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AD = 9. Let H be the rotation of point D by angle 5pi/6 around point A. Let B be the perpendicular bisector of segment HA. Line W is the reflection of line B across point D. Let K be the reflection of point A across point D. Point I is the reflection of point D across line W. Let U be the midpoint of segment KH. Find the measure of angle KIU, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let PK = 9. Let M be the rotation of point K by angle 11pi/12 around point P. Let J be the vector from point K to point P. Let N be the perpendicular bisector of segment MP. Point B is the reflection of point M across point K. Let G be point M translated by vector J. Line D is the perpendicular bisector of segment PB. Line F is the reflection of line D across line N. Point Z is the reflection of point B across line F. Point E is the midpoint of segment ZG. What is the length of segment PE? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ST = 3. Let A be the perpendicular bisector of segment TS. Point F lies on line A. Point L is the reflection of point S across point T. Point C is the reflection of point S across segment FL. What is the length of segment LC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let LA = 11. Point N is the reflection of point L across point A. Let F be the reflection of point A across point L. Let V be the perpendicular bisector of segment AF. Line J is the reflection of line V across point N. Point K is the reflection of point N across line J. What is 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 MA = 11. Let Z be the rotation of point M by angle pi/12 around point A. Let J be the reflection of point A across point Z. Let V be the midpoint of segment ZM. Let X be the perpendicular bisector of segment MV. Point F lies on line X at distance 3 from point M. Let W be the midpoint of segment FV. Point U is the reflection of point F across segment JZ. Point N is the reflection of point W across segment JZ. What is the length of segment UN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
DJ = 11. Let E be the reflection of point J across point D. Let V be the vector from point J to point E. Line G passes through point E perpendicular to segment DJ. Let S be the midpoint of segment DJ. Let X be the reflection of line G through point S. Point Y is on line G with |YE| = 1. Point F is the translation of point S by vector V. Point W is the translation of point D by vector V. Line Q is the perpendicular bisector of segment FY. Point N is the reflection of point W across point S. Line O is the reflection of line X across point Y. Let Z be the line through point W parallel to line G. Line U is the reflection of line Z across line O. Let C be the line through point D perpendicular to line Q. Point K is the reflection of point Y across line U. Point P is the midpoint of segment SN. Point L is the reflection of point P across line C. Find the distance between points K and L. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GF = 8. Point Z is obtained by rotating point G by angle pi/12 about point F. Let I be the perpendicular bisector of segment GF. Line M passes through point Z perpendicular to line I. Let D be the midpoint of segment GZ. Point U is the reflection of point F across line M. Point T is the reflection of point F across segment ZG. Let J be the midpoint of segment UD. What is the length of segment TJ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let KY = 7. Point M is the reflection of point Y across point K. Let B be the midpoint of segment KM. Let E be the reflection of point K across point M. Let P be the rotation of point K by angle pi/2 around point Y. Line F is the perpendicular bisector of segment KP. Point H is obtained by rotating point P by angle pi/3 about point K. Let J be the midpoint of segment MB. Point U is the reflection of point H across segment KE. Line V passes through points J and U. Let G be the reflection of line F across line V. Lines G and V intersect at point W. Find the distance between points K and W. Express your answer as a decimal number rounded to 2 decimal places.

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