[User]
Let YS = 5. Point J is obtained by rotating point Y by angle pi/12 about point S. Let Z be the reflection of point J across point S. Let F be the perpendicular bisector of segment JS. Point X is the reflection of point J across point Y. Let E be the perpendicular bisector of segment SY. Let I be the rotation of point S by angle pi/2 around point Z. Let O be the rotation of point J by angle pi/4 around point S. Point M is obtained by rotating point Z by angle pi/6 about point I. Point H is the midpoint of segment SX. Point A is obtained by rotating point M by angle pi about point J. Line G is the reflection of line F across line E. Point L is the intersection of line G and segment OA. Point D is the reflection of point M across segment SL. Find the length of segment HD. Express your answer as a decimal number rounded to 2 decimal places.

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


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JU = 12. Point M is the midpoint of segment UJ. Let Y be the line through point U perpendicular to segment UJ. Point E is the reflection of point M across line Y. What is the distance between points M 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 GH = 10. Line M is the perpendicular bisector of segment HG. Let D be the rotation of point G by angle pi/2 around point H. Point S is obtained by rotating point H by angle pi/2 about point G. Point C is obtained by rotating point S by angle 3pi/4 about point H. Let N be the line through point C parallel to line M. Line Z is the reflection of line N across line M. Let O be the rotation of point H by angle 11pi/12 around point G. Line W is the reflection of line N across point S. Point P is the reflection of point O across point G. Point Q is the reflection of point G across point P. Line T is the reflection of line Z across line W. Point K is the reflection of point S across line T. Point Y is the reflection of point Q across point K. Find the length of segment YD. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let PW = 12. Point Q is obtained by rotating point P by angle 5pi/12 about point W. Let K be the midpoint of segment QP. Let B be the rotation of point W by angle 2pi/3 around point P. Line O is the perpendicular bisector of segment KB. Line E passes through point K parallel to line O. Line U passes through point B perpendicular to line E. Point X is the reflection of point P across line U. Let D be the reflection of point X across point P. Point C is the reflection of point K across point D. What is the length of segment KC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LZ = 2. Let D be the vector from point Z to point L. Point F is the reflection of point Z across point L. Point H is the translation of point F by vector D. Point K is obtained by rotating point Z by angle pi/12 about point H. Let T be the midpoint of segment FK. Point A is the reflection of point H across point T. Point W is the reflection of point F across segment HK. Point Q is the reflection of point Z across point W. Find the length of segment QA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AH = 2. Point E is the reflection of point A across point H. Let R be the midpoint of segment HA. Let O be the rotation of point E by angle 3pi/4 around point R. Let D be the angle bisector of angle EOH. Let S be the reflection of line D through point R. Line I passes through points E and H. Lines I and S intersect at point U. Line F is the reflection of line S across point A. Point Z is the reflection of point U across line F. Point L is the midpoint of segment AZ. Let M be the midpoint of segment ZR. Find the distance between points M and L. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HT = 8. Let K be the rotation of point T by angle pi/4 around point H. Let E be the line through points H and K. Point D is the reflection of point T across point K. Point M is the reflection of point D across line E. Line F is the angle bisector of angle THM. Point N is the reflection of point M across line F. Line W is the perpendicular bisector of segment ND. Point P is the reflection of point H across point T. Point O is the reflection of point P across line W. Point X is the midpoint of segment OD. What is the length of segment OX? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let EW = 5. Let L be the midpoint of segment WE. Let S be the rotation of point W by angle 7pi/12 around point L. Point P is the reflection of point L across point W. Point N is the midpoint of segment EP. Point T is the midpoint of segment NS. Find the distance between points T and S. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
TW = 10. Line H passes through points W and T. Point K is obtained by rotating point W by angle 3pi/4 about point T. Let O be the angle bisector of angle TKW. Point J is the midpoint of segment WK. Point Q is the reflection of point J across line O. Line L is the reflection of line H across line O. Let P be the rotation of point T by angle 11pi/12 around point K. Let C be the reflection of line H across line L. Let E be the line through points K and T. Let A be the rotation of point Q by angle pi/6 around point K. Let F be the reflection of line O across line E. Point D is the reflection of point A across segment PW. Point V is the intersection of lines C and F. What is the length of segment VD? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KM = 12. L is constructed so that ML = 9. Let N be the perpendicular bisector of segment LM. Point D is the reflection of point M across point L. Point T lies on line N. Point X is the reflection of point D across point T. Point Z is the reflection of point X across line N. Find the distance between points Z and X. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IX = 4. Let T be the reflection of point I across point X. Point F is the reflection of point I across point T. Point M is the reflection of point I across point F. Let O be the midpoint of segment XT. Let G be the midpoint of segment XM. Point B is the midpoint of segment GT. Find the distance between points B 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 YX = 10. Let O be the reflection of point Y across point X. Let H be the rotation of point O by angle 5pi/12 around point X. Point B is the midpoint of segment YH. Let K be the midpoint of segment XY. Let C be the midpoint of segment KB. What is the length of segment KC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let FV = 8. Let Q be the rotation of point V by angle 5pi/6 around point F. Point N is the reflection of point V across point F. Point I is obtained by rotating point F by angle 3pi/4 about point V. Point T is the reflection of point F across segment VI. Let W be the midpoint of segment TN. Let O be the midpoint of segment FQ. Let R be the midpoint of segment OW. Let Z be the midpoint of segment VQ. Point A is the midpoint of segment ZR. Find the distance between points W and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RU = 6. Point I is obtained by rotating point R by angle pi/6 about point U. Point Q lies on segment IU. Point B is the reflection of point R across point Q. Line M is the perpendicular bisector of segment UR. Line T passes through point Q perpendicular to line M. Point H is the reflection of point I across line T. Point P is the reflection of point H across point B. Point W is the reflection of point U across point P. Point N is the midpoint of segment WI. What is the length of segment PN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JF = 5. Let M be the rotation of point J by angle pi/12 around point F. Point D is obtained by rotating point M by angle pi/12 about point F. Let H be the reflection of point F across point D. Point N is obtained by rotating point H by angle 11pi/12 about point F. Let L be the reflection of point D across point F. Point P is the reflection of point L across segment JF. Point G is the midpoint of segment PN. Point C is the midpoint of segment GH. What is the length of segment HC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let CU = 2. Let S be the perpendicular bisector of segment CU. Let F be the line through point C parallel to line S. Let L be the midpoint of segment UC. Point A is obtained by rotating point U by angle pi/6 about point L. Point E is obtained by rotating point A by angle 7pi/12 about point U. Point Z is obtained by rotating point A by angle pi/2 about point L. Point W is the reflection of point A across line F. Let H be the reflection of point A across point Z. Point K is the reflection of point H across segment UA. Let J be the reflection of point L across point E. Let T be the midpoint of segment JW. What is the length of segment TK? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let FY = 7. Let Q be the vector from point F to point Y. Point I is the midpoint of segment YF. Point A is the translation of point Y by vector Q. Let C be the midpoint of segment IF. Point K is the reflection of point C across point A. Point P is obtained by rotating point F by angle pi/12 about point A. Let V be the rotation of point Y by angle pi/12 around point P. Let U be the midpoint of segment VK. Point X is the midpoint of segment UA. Let N be the midpoint of segment IA. Let H be the midpoint of segment CN. Find the length of segment HX. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let MG = 9. Let Q be the perpendicular bisector of segment GM. Point I is the midpoint of segment GM. Line N passes through point M parallel to line Q. Point E is on line N with |EM| = 2. Line D is the angle bisector of angle EIM. Point Y is the reflection of point M across line D. Point C is the reflection of point E across point Y. Let J be the perpendicular bisector of segment IG. Let R be the reflection of line J across line N. Line T is the reflection of line R across line D. Let P be the reflection of point Y across point E. Point F is the midpoint of segment CM. Point X is the reflection of point F across segment CE. Line O passes through point P parallel to segment EG. Line V passes through point X parallel to line T. Let S be the midpoint of segment EG. Let U be the perpendicular bisector of segment EG. Line B passes through point I parallel to line U. Line H is the reflection of line O across line V. Lines B and H intersect at point A. What is the distance between points A and S? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IY = 5. Point M is the midpoint of segment IY. Point H is the reflection of point M across point Y. Point S is the reflection of point Y across point H. Let J be the reflection of point M across point I. What is the distance between points J and S? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BW = 9. Point N is the reflection of point W across point B. Let H be the rotation of point W by angle 11pi/12 around point B. Point I is obtained by rotating point W by angle pi/3 about point N. Point J is the midpoint of segment IN. Point U is the midpoint of segment HB. Point L is the reflection of point I across segment JU. What is the length of segment LN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BI = 11. Point V is obtained by rotating point B by angle 5pi/6 about point I. Let L be the angle bisector of angle IVB. Point P is the reflection of point V across point B. Let W be the midpoint of segment BP. Line Q passes through point W perpendicular to line L. Point K is the reflection of point V across line Q. Find the distance between points P and K. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WJ = 4. Line M is the perpendicular bisector of segment JW. Let P be the rotation of point W by angle 7pi/12 around point J. Line E is the reflection of line M across point P. Let K be the line through points P and W. Let U be the reflection of line M across line K. Point O is the intersection of lines U and K. Lines E and U intersect at point I. Line X is the angle bisector of angle JOP. Let G be the midpoint of segment JP. Let L be the midpoint of segment IP. Let Q be the midpoint of segment GP. Point F is the reflection of point Q across line X. Point B is the reflection of point G across point L. Point T is the midpoint of segment BF. Find the distance between points T and L. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ZR = 1. Let K be the rotation of point R by angle 11pi/12 around point Z. Line T passes through points Z and K. Point Y is the midpoint of segment KR. Let N be the line through point Z perpendicular to segment ZR. Line F is the reflection of line N across line T. Let H be the reflection of point Y across point Z. Point O is the reflection of point H across line F. Point X is the reflection of point H across segment ZR. What is the length of segment XO? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GU = 10. Let Q be the line through points G and U. Let X be the rotation of point U by angle pi/4 around point G. Point A is on line Q. Let I be the line through point U perpendicular to line Q. Let Z be the rotation of point A by angle 5pi/12 around point U. Point R is the reflection of point X across line I. Let D be the midpoint of segment RU. Let Y be the midpoint of segment RD. Let F be the reflection of point D across point U. Let T be the midpoint of segment ZY. Let N be the reflection of point T across point F. Point C is the reflection of point T across point R. What is the distance between points N 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 KI = 1. Line A is the perpendicular bisector of segment KI. Let E be the reflection of point K across point I. Point T is obtained by rotating point K by angle 3pi/4 about point E. Point X is the reflection of point T across segment KI. Point L is the midpoint of segment KX. Point O is the reflection of point L across line A. Point W is obtained by rotating point K by angle 5pi/6 about point X. Point B is the reflection of point L across segment EX. Let M be the line through points B and L. Point Q is the intersection of line M and segment WE. Point R is the midpoint of segment QO. What is the length of segment RE? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let CX = 9. Point I is obtained by rotating point X by angle pi about point C. Let Y be the rotation of point I by angle pi/3 around point X. Point E is the midpoint of segment XC. Point B is the reflection of point X across point Y. Let A be the reflection of point E across point X. Line P is the perpendicular bisector of segment IC. Let J be the reflection of line P through point B. Line S passes through points A and X. Let Q be the reflection of point A across point B. Point H is the intersection of lines J and S. What is the length of segment HQ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HV = 7. Point F is the midpoint of segment HV. Let C be the rotation of point V by angle 5pi/12 around point F. Let I be the line through points C and F. Point J is obtained by rotating point V by angle pi/4 about point C. Line T is the reflection of line I across point V. Point G is the midpoint of segment VF. Let B be the line through points G and J. Line M is the reflection of line T across line B. Point P is the reflection of point H across line B. Point X is the reflection of point H across line M. Find the length of segment XP. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LQ = 5. Point S is obtained by rotating point Q by angle pi about point L. Point E is obtained by rotating point S by angle 11pi/12 about point L. Point V is obtained by rotating point S by angle 5pi/6 about point Q. Let B be the midpoint of segment VE. Point R is obtained by rotating point V by angle 11pi/12 about point B. Point Z is the reflection of point E across segment QL. Point H is the reflection of point B across segment RZ. Let I be the midpoint of segment HB. Find the distance between points R and I. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ZA = 6. Let T be the rotation of point A by angle 11pi/12 around point Z. Point J is the midpoint of segment ZA. Let O be the rotation of point T by angle 5pi/12 around point Z. Let S be the midpoint of segment JO. Let D be the midpoint of segment ZO. Point C is the reflection of point A across segment ZO. Let E be the midpoint of segment DZ. Let G be the reflection of point C across point E. Find the length of segment SG. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LT = 3. Line C is the perpendicular bisector of segment TL. Point Z is on line C with |ZL| = 7. Let M be the line through point L parallel to line C. Line U is the reflection of line M across point T. Point F is the reflection of point Z across line U. Let X be the line through point F parallel to line U. Point W is the reflection of point T across line X. What is the length of segment WF? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
MT = 1. Let Z be the perpendicular bisector of segment TM. Let C be the rotation of point M by angle 3pi/4 around point T. Let V be the midpoint of segment CT. Point E is the reflection of point C across line Z. Point K is the midpoint of segment VE. Point W is the reflection of point M across point K. Point P is the reflection of point K across point C. What is the length of segment PW? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let EO = 7. Point X is the reflection of point E across point O. Line A is the perpendicular bisector of segment XO. Let N be the reflection of point O across point E. Point F is the reflection of point E across line A. Point J is obtained by rotating point X by angle 7pi/12 about point F. Let Q be the rotation of point E by angle pi/6 around point O. Let K be the angle bisector of angle JEQ. Point L is the reflection of point Q across line K. Point R is the reflection of point N across point Q. Point U is the reflection of point L across segment JN. 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}


[User]
MI = 4. Let B be the midpoint of segment MI. Point E is the reflection of point B across point I. Let K be the angle bisector of angle MBI. Point L is the midpoint of segment BI. Point X is obtained by rotating point M by angle pi/3 about point L. Line Q is the perpendicular bisector of segment LX. Line R is the reflection of line Q across line K. Point H is the intersection of segment LX and line R. What is 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}


[User]
Let JU = 6. Point R is obtained by rotating point J by angle 5pi/12 about point U. Point F is the reflection of point J across point U. Point G is the midpoint of segment RF. Let H be the angle bisector of angle JGF. Let C be the line through point U perpendicular to line H. Let D be the line through point F perpendicular to line C. Point Q is the midpoint of segment UJ. Point T is the reflection of point Q across line D. What is the distance between points T and G? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OI = 4. Let A be the rotation of point O by angle 5pi/12 around point I. Point F is the midpoint of segment OA. Point Q is the reflection of point O across segment AI. Let L be the line through point I perpendicular to segment OF. Point X is the reflection of point Q across line L. Let R be the reflection of point Q across point X. Point B is the reflection of point R across segment OF. Find the distance between points B and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let XG = 12. Let Z be the midpoint of segment XG. Let A be the rotation of point Z by angle 7pi/12 around point X. Point I is the reflection of point A across point G. Point H is the reflection of point Z across point A. Line P is the angle bisector of angle HGX. Point R is the reflection of point I across line P. Line J is the perpendicular bisector of segment AZ. Let F be the reflection of line P across line J. Point W is the reflection of point R across line F. Line M is the angle bisector of angle IHX. Point D is the reflection of point Z across line P. Point K is the reflection of point W across line M. What is the distance between points D and K? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
TD = 7. Let N be the reflection of point T across point D. Let Y be the rotation of point N by angle 5pi/12 around point D. Let V be the reflection of point D across point Y. Let S be the line through points N and V. Let Z be the line through point Y parallel to line S. Point P is the reflection of point N across line Z. What is the length of segment VP? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RT = 9. Point K is obtained by rotating point R by angle 7pi/12 about point T. Point V is obtained by rotating point K by angle 5pi/6 about point T. Let F be the rotation of point K by angle pi around point T. Point Q is the midpoint of segment VT. Let W be the midpoint of segment TR. Let L be the line through point W perpendicular to segment TR. Point N is the reflection of point F across line L. Point O is the midpoint of segment NT. What is the measure of angle WOQ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let XF = 2. Let U be the midpoint of segment XF. Point K is obtained by rotating point F by angle 7pi/12 about point U. Line T passes through points F and K. Point M is the midpoint of segment XK. Point V is the reflection of point M across line T. Let P be the midpoint of segment UM. Find the distance between points V and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LX = 7. Let C be the perpendicular bisector of segment LX. Point A is on line C with |AL| = 8. Let W be the angle bisector of angle XLA. Point T is the reflection of point X across line W. Point D is the midpoint of segment TA. Let Q be the vector from point A to point D. Point P lies on line W. Let J be the angle bisector of angle DAT. Point F is the midpoint of segment TL. Point E is the translation of point P by vector Q. Point O is the midpoint of segment XP. Let Y be the line through point E parallel to line J. Line I is the reflection of line W across line Y. Line Z is the reflection of line W across point A. Point G is the reflection of point F across point O. Point H is the intersection of lines I and Z. What is the distance between points G and H? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
UQ = 9. Point D is the reflection of point Q across point U. Line N is the perpendicular bisector of segment DQ. Let V be the vector from point Q to point D. Let J be the line through point Q parallel to line N. Let S be the midpoint of segment QU. Line K is the perpendicular bisector of segment QS. Point M is the midpoint of segment UD. Let C be point M translated by vector V. Point A is on line N with |AQ| = 6. Point W is the midpoint of segment UC. Line Y passes through point W parallel to line K. Line X passes through point A perpendicular to line Y. Let L be the reflection of line X through point Q. Point G is the reflection of point D across line L. Point P is the reflection of point A across line J. Find the distance between points P and G. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KY = 4. Let B be the rotation of point Y by angle pi/12 around point K. Point R is the midpoint of segment KY. Point P is the reflection of point B across point K. Let M be the reflection of point P across point R. Let D be the reflection of point B across point R. Let X be the reflection of point D across point P. Let A be the angle bisector of angle DRX. Line J is the reflection of line A across point K. Point U is the reflection of point K across point M. Point C is the reflection of point R across line J. Let E be the midpoint of segment UB. Let L be the line through points C and P. Point G is obtained by rotating point M by angle 7pi/12 about point D. Line N is the perpendicular bisector of segment YP. Point T is the reflection of point C across line N. Point O is the reflection of point G across point T. Point V is the intersection of segment ET and line L. What is the distance between points O and V? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
CA = 2. Point G is the midpoint of segment AC. Point T is obtained by rotating point G by angle 2pi/3 about point A. Let Z be the midpoint of segment TG. Point U is the reflection of point Z across point C. Let L be the midpoint of segment TU. What is 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]
Let CH = 1. Line P is the perpendicular bisector of segment HC. Point N lies on line P at distance 10 from point C. Let M be the midpoint of segment NH. Let D be the reflection of line P through point H. Let B be the reflection of line D through point M. Let U be the perpendicular bisector of segment MC. Line J passes through point M perpendicular to line B. Point W is the reflection of point N across point H. Point O is the reflection of point W across line J. Point E is the reflection of point N across line J. Point F is the midpoint of segment WO. Point L is the reflection of point F across segment NH. Point Z is the reflection of point E across line U. What is the distance between points L 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 DG = 1. Let P be the rotation of point D by angle pi/12 around point G. Let U be the midpoint of segment DP. Let Y be the midpoint of segment DG. Let Q be the line through points Y and G. Point W is the reflection of point U across line Q. What is the measure of angle YWU, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WJ = 5. Point T is the midpoint of segment WJ. Line I is the perpendicular bisector of segment JW. Let P be the rotation of point T by angle 7pi/12 around point W. Let Q be the midpoint of segment JT. Point Z is the midpoint of segment QP. Line R passes through point Z parallel to line I. Let F be the reflection of point Z across point W. Line R and segment JW intersect at point K. Find the length of segment FK. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YD = 9. Let S be the rotation of point Y by angle pi/12 around point D. Point F is the midpoint of segment YS. Let Z be the perpendicular bisector of segment YF. Point V is obtained by rotating point D by angle pi/6 about point F. Point Q is the reflection of point F across point Y. Line H passes through point V parallel to line Z. Point U is the reflection of point Q across line H. Let C be the reflection of point F across point V. Find the distance between points U 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 BC = 6. Let G be the vector from point C to point B. Let O be the midpoint of segment BC. Let V be point B translated by vector G. Point M is obtained by rotating point V by angle 11pi/12 about point O. Point P is obtained by rotating point C by angle 5pi/12 about point V. Let Q be the reflection of point M across point O. Point I is obtained by rotating point Q by angle 3pi/4 about point B. Let W be the reflection of point I across point B. Line S is the perpendicular bisector of segment PO. Point X is the reflection of point V across point W. Let L be the reflection of line S through point V. Point D is the reflection of point V across line L. Line N passes through point D parallel to line L. Point F is the intersection of line N and segment BC. Point R is the reflection of point W across segment BC. Point E is the midpoint of segment XF. Find the length of segment RE. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VD = 6. Point A is the midpoint of segment VD. Point S is obtained by rotating point V by angle pi/6 about point A. Point F is the reflection of point S across segment VD. Point M is the midpoint of segment VS. Let T be the midpoint of segment MF. Find the length of segment TS. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VJ = 11. Line A passes through points V and J. Point Z is obtained by rotating point J by angle 5pi/12 about point V. Point U is the reflection of point Z across line A. Line W passes through points Z and J. Let B be the rotation of point Z by angle 7pi/12 around point U. Let X be the perpendicular bisector of segment VU. Let I be the reflection of line W across line X. Point Y is the reflection of point U across line I. What is the length of segment BY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LO = 2. Line Y is the perpendicular bisector of segment OL. Let E be the rotation of point L by angle 5pi/6 around point O. Point D is obtained by rotating point L by angle pi/2 about point O. Point G is the reflection of point D across point L. Point F is the midpoint of segment DO. Point I is the reflection of point D across line Y. Let J be the perpendicular bisector of segment FE. Let M be the reflection of point D across point I. Line H is the reflection of line J across point G. Point T is the midpoint of segment IG. Point Q is the reflection of point M across line H. 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]
HL = 4. Let I be the line through points H and L. Let X be the line through point L perpendicular to line I. Point M is obtained by rotating point L by angle 11pi/12 about point H. Line B passes through point M parallel to line X. Point R is the reflection of point H across point M. Lines I and B intersect at point Q. Point F is the reflection of point R across point Q. What is the length of segment RF? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let MJ = 10. Point N is the midpoint of segment MJ. Let I be the reflection of point J across point M. Let V be the rotation of point J by angle pi/6 around point N. Let T be the midpoint of segment NJ. Point Q is the midpoint of segment VI. Point Z is the reflection of point N across segment VI. Point R is the midpoint of segment TZ. Let L be the reflection of point V across point R. Find the length of segment LQ. 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 ZM = 1. B is constructed so that MB = 1. Point H is the reflection of point M across point Z. Line X passes through points H and B. C is placed 1 units away from B. Point I is the reflection of point Z across line X. Point F is the midpoint of segment IC. Let A be the midpoint of segment CH. Find the distance between points F and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FU = 11. Let J be the rotation of point U by angle 11pi/12 around point F. Line N passes through points U and F. Point E is obtained by rotating point F by angle 3pi/4 about point J. Line Q passes through points E and U. Line Z is the reflection of line N across line Q. Let O be the reflection of line Q across line Z. Point R is the reflection of point F across point E. Point T is the reflection of point J across line O. What is the length of segment TR? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WG = 11. Point S is the reflection of point G across point W. C is placed 11 units away from G. Let B be the reflection of point S across point G. Let Z be the reflection of point S across point B. Point O is the midpoint of segment ZC. Point E is the midpoint of segment CW. What is the distance between points O 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 ZI = 5. Point V is the reflection of point Z across point I. Let A be the reflection of point I across point V. Let E be the midpoint of segment IV. What is the distance between points A and E? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let UY = 7. Let O be the reflection of point Y across point U. Let G be the rotation of point O by angle 7pi/12 around point U. Let M be the midpoint of segment UG. Let W be the reflection of point O across point G. Point K is the midpoint of segment WG. Find the distance between points K 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 TX = 6. Point P is obtained by rotating point T by angle 5pi/6 about point X. Point J is the midpoint of segment XT. Let V be the midpoint of segment PX. Point K is the midpoint of segment VT. Let F be the midpoint of segment KT. Line L passes through point F perpendicular to segment PX. Let U be the line through point J perpendicular to line L. Point D is the midpoint of segment JT. Point Y is the reflection of point D across line U. Find the length of segment YP. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let UB = 1. Point J is obtained by rotating point U by angle pi/4 about point B. Let Z be the reflection of point J across point U. Point V is the midpoint of segment BU. Line X is the angle bisector of angle ZBJ. Point H is the reflection of point V across line X. What is the distance between points H and Z? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OG = 5. Let V be the rotation of point G by angle pi around point O. Let M be the vector from point G to point V. Point W is the translation of point O by vector M. Let P be the rotation of point V by angle 5pi/6 around point G. Line D passes through points W and G. Point U is the midpoint of segment OG. Line E passes through point V perpendicular to line D. Line K is the reflection of line E across point U. Line H passes through point O parallel to line K. Lines K and D intersect at point F. Line N passes through points G and P. Point I is the intersection of lines H and N. Point J is the midpoint of segment IF. 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 QX = 6. Point V is the reflection of point X across point Q. Let Y be the angle bisector of angle XQV. Line B passes through point V perpendicular to line Y. Let E be the line through point V parallel to line Y. Let S be the vector from point X to point V. Line J is the perpendicular bisector of segment VQ. Line O is the reflection of line E across line Y. Point I is the translation of point V by vector S. Point H is on line O. Let D be the reflection of line B through point H. Let F be the reflection of point V across point I. Let C be the midpoint of segment FX. Point K is the intersection of lines E and D. Line U is the reflection of line J across line Y. Lines U and D intersect at point M. Let R be the midpoint of segment CM. Let N be the midpoint of segment KF. What is the length of segment RN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NY = 11. Let L be the reflection of point Y across point N. Line A passes through points Y and L. Let U be the reflection of point Y across point L. Let X be the angle bisector of angle UNY. Point P is obtained by rotating point L by angle 5pi/12 about point Y. Line G passes through point P perpendicular to line X. Let S be the rotation of point U by angle pi/12 around point P. Point F is the reflection of point Y across point U. Let E be the midpoint of segment FU. Point O is the reflection of point Y across line G. Point C is obtained by rotating point N by angle pi/6 about point L. Point B is the midpoint of segment EC. Line W is the angle bisector of angle SBU. Point Z is the reflection of point Y across line W. Let R be the midpoint of segment ON. Point T is the reflection of point R across line A. Find the distance between points T 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 EY = 4. Point Q is the reflection of point E across point Y. Let B be the rotation of point Q by angle 5pi/12 around point E. Point O is the midpoint of segment YQ. Point R is the midpoint of segment YO. Point X is the reflection of point R across segment YB. Let C be the line through point O parallel to segment YB. Point D is the reflection of point X across line C. Point F is the reflection of point R across point D. Find the distance between points X and F. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XE = 7. Let H be the rotation of point E by angle 11pi/12 around point X. Point W is obtained by rotating point X by angle pi about point H. Point K is the reflection of point X across point W. Point A is the reflection of point E across point W. Let O be the line through points A and W. Point C is the reflection of point K across line O. Find the distance between points K and C. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XJ = 4. Let C be the line through point J perpendicular to segment XJ. Let S be the rotation of point X by angle 5pi/12 around point J. Let Z be the rotation of point J by angle pi/6 around point X. Let R be the reflection of line C through point Z. Let P be the reflection of line C across line R. Let N be the line through point S perpendicular to line R. Line P and segment XJ intersect at point Q. Let W be the reflection of point Q across point Z. Point H is the reflection of point Q across line N. What is 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 FY = 6. Let A be the rotation of point Y by angle pi/12 around point F. Point M is the midpoint of segment FY. Let R be the reflection of point A across point M. Point U is the midpoint of segment MR. Point Z is the reflection of point F across point U. What is the length of segment ZA? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KV = 12. Point Y is obtained by rotating point V by angle pi/2 about point K. Let P be the midpoint of segment YV. Let N be the midpoint of segment PK. Let W be the perpendicular bisector of segment KY. Let A be the line through point K parallel to line W. Point C is obtained by rotating point P by angle pi/6 about point V. Point T is the reflection of point Y across line A. Let B be the reflection of point T across point N. Let F be the angle bisector of angle TCN. Point I is the midpoint of segment YK. Let L be the reflection of line F through point I. Point E is the intersection of segment YK and line L. Let G be the midpoint of segment EI. Point U is the midpoint of segment YG. What is the distance between points U 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 ZB = 2. Point L is obtained by rotating point Z by angle 7pi/12 about point B. Let K be the midpoint of segment BL. Point S is the reflection of point K across segment LZ. Let F be the midpoint of segment ZK. Point V is the reflection of point K across segment SB. What is the distance between points V and F? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IY = 7. Let K be the vector from point I to point Y. Let A be point Y translated by vector K. Point M is obtained by rotating point Y by angle 5pi/6 about point I. Line E is the perpendicular bisector of segment YA. Point B is obtained by rotating point I by angle 3pi/4 about point Y. Let P be the midpoint of segment MA. Point Q is the reflection of point P across line E. Point V is the midpoint of segment BM. Find the distance between points Q and V. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BI = 1. Let E be the vector from point B to point I. Let H be the reflection of point I across point B. Point V is the translation of point I by vector E. Line R is the perpendicular bisector of segment BH. Point Y is on line R with |YI| = 11. Let J be the reflection of line R through point V. Line D passes through point Y parallel to segment BH. Point N is the reflection of point Y across segment BH. Let M be the line through points V and Y. Point A is the reflection of point V across line D. Let F be the reflection of line M across line J. Point L is the reflection of point N across line F. Find the length of segment AL. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HK = 4. Point Y is the reflection of point K across point H. Point I is the reflection of point K across point Y. Line L is the perpendicular bisector of segment KI. Let C be the reflection of point Y across point K. Point S lies on line L at distance 5 from point Y. Point O is the midpoint of segment CS. Point P is the reflection of point C across line L. Point U is the midpoint of segment YO. Point R is the reflection of point U across segment SK. Find the distance between points P and R. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AV = 10. Point Y is the reflection of point A across point V. Point N is obtained by rotating point A by angle pi/2 about point V. Let K be the line through point Y perpendicular to segment AN. Point R is the reflection of point V across line K. Point S is the reflection of point R across segment AN. What is the measure of angle SVR, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VA = 9. Point J is the reflection of point A across point V. Let P be the rotation of point A by angle pi/6 around point J. Line D is the perpendicular bisector of segment JA. Let W be the reflection of point V across point A. Let O be the rotation of point W by angle 3pi/4 around point J. Point K is the midpoint of segment PO. Let Z be the reflection of line D through point K. Line T is the reflection of line Z across point J. Point I is the intersection of line T and segment JW. Find the length of segment PI. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let QX = 7. Point T is the reflection of point X across point Q. Point I is the reflection of point Q across point X. Point W is the midpoint of segment QT. Point B is the reflection of point X across point I. Let V be the rotation of point B by angle pi/3 around point Q. Let C be the rotation of point X by angle pi/6 around point I. Point J is obtained by rotating point I by angle pi/2 about point W. Point L is the midpoint of segment WV. Point P is the reflection of point J across segment WV. Point Y is the reflection of point B across segment WV. Let A be the midpoint of segment LQ. Point F is the reflection of point Y across segment CP. Find the distance between points F and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YV = 11. Point K is obtained by rotating point Y by angle pi/3 about point V. Let F be the reflection of point Y across point V. Line I passes through points F and Y. Let A be the reflection of line I through point K. Point B is the reflection of point K across line I. Point O is the reflection of point B across point V. Let L be the perpendicular bisector of segment VB. Let E be the reflection of line L across line A. Line P is the reflection of line E across point O. Point S is the reflection of point B across line P. What is the distance between points Y and S? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BI = 2. Point M is the midpoint of segment BI. Point Y is the midpoint of segment MI. Let S be the rotation of point Y by angle pi/2 around point M. Let P be the rotation of point S by angle pi/12 around point B. Point F is obtained by rotating point Y by angle pi about point P. Let R be the line through points P and S. Let T be the line through points Y and P. Point W is the reflection of point B across line T. Let X be the reflection of line R through point W. Point C is the midpoint of segment PM. Point A is the reflection of point F across line X. What is the distance between points A 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 EC = 6. Let X be the reflection of point E across point C. Point R is obtained by rotating point X by angle 3pi/4 about point E. Point U is the reflection of point R across point C. Line Y passes through points E and R. Point V is the reflection of point U across line Y. Let T be the midpoint of segment XR. Let Q be the line through points X and T. Point G is the reflection of point V across line Q. What is the length of segment EG? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PT = 1. Let X be the rotation of point P by angle 5pi/12 around point T. Point I is the reflection of point T across point X. Let A be the reflection of point P across point T. Line R is the angle bisector of angle IXA. Line U passes through points I and P. Line N passes through point T parallel to line R. Point M is obtained by rotating point A by angle 7pi/12 about point I. Line G passes through point T parallel to line U. Line O is the reflection of line G across line N. Let E be the reflection of line N through point M. Lines E and O intersect at point W. Point K is the midpoint of segment AW. Let J be the midpoint of segment IA. What is the length of segment JK? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VC = 7. Let O be the midpoint of segment VC. Let X be the reflection of point O across point V. Point Z is the midpoint of segment VX. Point D is the reflection of point Z across point O. Find the distance between points D and V. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
DO = 8. Line Q is the perpendicular bisector of segment OD. Point I lies on line Q at distance 3 from point D. Line S passes through point O parallel to line Q. Let F be the vector from point I to point O. Point A is the translation of point D by vector F. Point H is the reflection of point D across point A. Line N is the perpendicular bisector of segment IH. Let U be the reflection of line S across line N. Let B be the reflection of line N through point D. Point J is the intersection of lines B and U. What is the length of segment AJ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NX = 1. Let U be the vector from point N to point X. Let E be the rotation of point X by angle 7pi/12 around point N. Point B is obtained by rotating point E by angle 7pi/12 about point X. Let I be the rotation of point E by angle 7pi/12 around point B. Let H be point E translated by vector U. Let Z be the vector from point N to point I. Point S is the translation of point H by vector Z. Point W is the midpoint of segment SH. Line A passes through points X and N. Point V is the intersection of segment IH and line A. Point G is the reflection of point W across line A. Point P is obtained by rotating point V by angle pi/6 about point H. Line R passes through points X and W. Point D is the reflection of point G across line R. Let L be the midpoint of segment PG. What is the length of segment LD? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let JR = 7. Let H be the line through points J and R. Point B is obtained by rotating point R by angle pi/6 about point J. Point G is the midpoint of segment BR. Line F is the perpendicular bisector of segment JG. Let T be the reflection of line F across line H. Line D is the reflection of line H across line T. Point I is the intersection of lines T and D. Line D and segment JB intersect at point Z. What is the distance between points Z and I? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LX = 9. Let P be the vector from point X to point L. Let Q be point L translated by vector P. Let S be the rotation of point Q by angle 11pi/12 around point X. Let T be the perpendicular bisector of segment QL. Let G be the midpoint of segment XL. Point E is the reflection of point S across line T. Find the distance between points E 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 EN = 3. Let V be the rotation of point N by angle pi/3 around point E. Let F be the rotation of point V by angle 3pi/4 around point E. Let H be the rotation of point E by angle pi/12 around point F. Line I passes through points H and E. Point G is the reflection of point E across point H. Line R passes through point F parallel to line I. Line D is the reflection of line I across line R. Point Z is the reflection of point G across line D. Point K is the midpoint of segment NZ. What is the distance between points Z 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 TD = 8. Let J be the midpoint of segment TD. Let C be the perpendicular bisector of segment JT. Point U is on line C with |UJ| = 10. Point Q is the midpoint of segment DJ. Let I be the perpendicular bisector of segment QT. Line K is the reflection of line I across point U. Point V is obtained by rotating point Q by angle pi/4 about point D. Segment VT and line K intersect at point A. Let P be the midpoint of segment JA. Point N is the reflection of point Q across point P. Point M is the midpoint of segment PJ. Find the length of segment NM. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BE = 11. Let N be the line through points B and E. Point S is obtained by rotating point E by angle pi/2 about point B. Point T is the reflection of point S across line N. Point X lies on line N at distance 1 from point S. Point M is the midpoint of segment XT. Let L be the midpoint of segment MB. Let A be the midpoint of segment SL. Point D is the intersection of segments AT and MB. Find the distance between points D and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HP = 10. Point C is the reflection of point H across point P. Point L is obtained by rotating point P by angle 5pi/6 about point C. Point G is obtained by rotating point H by angle 7pi/12 about point L. Line B passes through points G and H. Point V is the reflection of point G across point C. Point K is the reflection of point C across line B. Let U be the reflection of point K across point V. Find the distance between points U 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 MV = 1. Point F is the midpoint of segment MV. Let E be the angle bisector of angle MFV. Point U is on line E. Point B is the midpoint of segment MF. Let K be the midpoint of segment UV. Let C be the midpoint of segment BU. What is the distance between points K and C? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BV = 8. Let Q be the vector from point B to point V. Let X be the vector from point B to point V. Point G is the midpoint of segment BV. Let O be the rotation of point G by angle pi/12 around point V. Let M be point O translated by vector X. Point H is the translation of point V by vector Q. Point E is the reflection of point M across segment BV. Point Y is the reflection of point M across segment EG. Point U is the midpoint of segment EG. Point J is the reflection of point H across point U. What is the distance between points J 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 FX = 2. Let K be the reflection of point X across point F. Point U is the midpoint of segment FX. Let S be the reflection of point X across point K. Point Q is obtained by rotating point K by angle pi/6 about point U. Let T be the midpoint of segment QU. Line J is the perpendicular bisector of segment FS. Let Y be the line through point T parallel to line J. Point L is the reflection of point F across point S. Point Z is the reflection of point L across line Y. Point A is the reflection of point T across segment FX. Find the length of segment ZA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RF = 11. Let W be the rotation of point R by angle 7pi/12 around point F. Let T be the midpoint of segment RW. Let J be the midpoint of segment RT. Let RL = 8. Line X is the angle bisector of angle RJL. Line Z is the perpendicular bisector of segment WJ. Point D is the reflection of point T across line Z. Point C is the reflection of point D across line X. Find the length of segment JC. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GC = 9. Let Y be the rotation of point G by angle pi around point C. Line U is the perpendicular bisector of segment YC. Point N lies on segment CY. Point I lies on line U at distance 8 from point G. Point A is the reflection of point G across line U. Let Q be the reflection of point I across point N. Let J be the angle bisector of angle GNY. Point V is the reflection of point Q across line J. What is the length of segment AV? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WD = 8. Line N is the perpendicular bisector of segment WD. Point A is the midpoint of segment WD. Point U is the midpoint of segment DA. Line F is the reflection of line N across point W. Let G be the reflection of point U across point D. Let T be the rotation of point U by angle 11pi/12 around point W. Point I is the midpoint of segment UT. Let R be the reflection of point U across point G. Point Z is the midpoint of segment IR. Let V be the reflection of line N across line F. Line O passes through point Z parallel to segment TA. Point L is the intersection of lines V and O. Find the distance between points U 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 PD = 5. Point V is obtained by rotating point P by angle 5pi/6 about point D. Let I be the midpoint of segment VP. Point S is the midpoint of segment VD. Point O is the reflection of point D across point I. Find the distance between points S 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 JE = 3. Let X be the rotation of point E by angle 3pi/4 around point J. Let B be the rotation of point X by angle 3pi/4 around point E. Let D be the reflection of point X across point B. Point T is the reflection of point J across point B. Let R be the reflection of point T across point J. Let K be the perpendicular bisector of segment XR. Let S be the reflection of line K through point E. Point O is the reflection of point D across line S. Find the distance between points O and R. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NI = 10. Let H be the midpoint of segment NI. Line C is the perpendicular bisector of segment IN. Point P is obtained by rotating point H by angle pi/2 about point N. Point W is obtained by rotating point H by angle pi/3 about point N. Point X is the reflection of point N across point P. Let Y be the rotation of point X by angle 3pi/4 around point W. Let U be the perpendicular bisector of segment WH. Line O is the reflection of line U across point X. Let A be the line through point P parallel to line O. Let V be the line through point X perpendicular to line C. Point Z is the reflection of point X across line A. Line K is the reflection of line V across point Z. Point B is the reflection of point Y across line K. Let T be the midpoint of segment ZB. What is the distance between points T and X? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let DF = 6. Let X be the midpoint of segment FD. Point Z is obtained by rotating point X by angle pi/12 about point D. Let K be the reflection of point X across point D. Point B is the reflection of point D across segment XZ. Line L is the perpendicular bisector of segment KD. Point I is the reflection of point B across line L. Find the measure of angle XBI, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
XO = 3. Point W is the reflection of point X across point O. Point A is the reflection of point X across point W. Let P be the rotation of point O by angle 7pi/12 around point W. Line D passes through point A perpendicular to segment XW. Let B be the reflection of point P across point W. Point H is obtained by rotating point X by angle pi/6 about point O. Let L be the perpendicular bisector of segment HW. Let Q be the reflection of line L through point P. Let G be the reflection of point X across point B. Point I is the reflection of point G across segment XW. Line C is the reflection of line D across line Q. Point K is the intersection of segment XW and line C. What is the distance between points I 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 JS = 4. Let B be the rotation of point J by angle pi/3 around point S. Let C be the midpoint of segment JS. Let G be the rotation of point B by angle pi/4 around point J. Let L be the rotation of point B by angle 5pi/12 around point C. Let F be the perpendicular bisector of segment BG. Let W be the line through points J and L. Let H be the reflection of line F across line W. Point V is the midpoint of segment SC. Point P is the midpoint of segment VJ. Point E is the reflection of point B across line H. Find 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}


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YP = 2. Let M be the reflection of point Y across point P. Point R is the midpoint of segment YP. Let Q be the rotation of point Y by angle 3pi/4 around point R. Let V be the perpendicular bisector of segment QP. Line U is the perpendicular bisector of segment MQ. Let L be the midpoint of segment QY. Let N be the reflection of line U across line V. Line Z is the reflection of line U across line N. Let K be the rotation of point Q by angle 3pi/4 around point L. Line H passes through point K parallel to line Z. Point T is the intersection of lines H and N. Point W is the reflection of point T across line Z. Point I is the midpoint of segment MR. Find the length of segment WI. 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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CM = 2. Let U be the line through points M and C. Let X be the rotation of point M by angle pi/6 around point C. Let B be the reflection of line U through point X. Point N is the reflection of point X across line B. Let W be the line through points M and X. Lines B and W intersect at point R. 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}


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Let LD = 2. Let F be the line through points L and D. Let M be the reflection of point L across point D. Point I is on line F. Point H is on line F with |HI| = 8. Point S lies on line F at distance 10 from point I. Let Y be the perpendicular bisector of segment SD. Point G is on line Y with |GH| = 8. Let R be the reflection of point G across point H. Let U be the midpoint of segment HR. Point T is the midpoint of segment MU. Point Q is the midpoint of segment UL. 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}


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JB = 6. Point S is obtained by rotating point J by angle pi/3 about point B. Let E be the rotation of point S by angle pi/4 around point B. Let U be the midpoint of segment SJ. Line F passes through point J perpendicular to segment UB. Point N is the reflection of point E across line F. Line V passes through points B and J. Point L is the midpoint of segment UB. Point Y is the reflection of point N across segment UB. Let I be the reflection of point E across point N. Point D is the reflection of point L across line V. Line K is the angle bisector of angle LJI. Let G be the midpoint of segment NY. Point Q is the reflection of point D across line K. Find the length of segment GQ. Express your answer as a decimal number rounded to 2 decimal places.

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


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LY = 1. Point P is obtained by rotating point L by angle pi about point Y. Let E be the rotation of point Y by angle 3pi/4 around point P. Let G be the line through points E and L. Let F be the line through point Y parallel to line G. Let I be the line through points L and P. Point J is the reflection of point L across segment PE. Let B be the line through point J parallel to line G. Point M is the midpoint of segment PE. Let H be the midpoint of segment EM. Let K be the reflection of line F across line B. Point R is the reflection of point E across line K. Point W is the reflection of point R across segment PE. Let T be the midpoint of segment WH. Point O is the intersection of lines I and K. What is the length of segment OT? 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 QW = 1. Let V be the midpoint of segment WQ. Point N is obtained by rotating point Q by angle pi/4 about point W. Let M be the midpoint of segment NV. Let R be the rotation of point W by angle 11pi/12 around point N. Point B is the reflection of point N across point R. Point C is the reflection of point Q across point R. Point U is the midpoint of segment CB. Line T is the angle bisector of angle RMV. Point G is the reflection of point U across line T. Point F is the midpoint of segment WG. 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}


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GS = 8. Let K be the reflection of point G across point S. Let D be the midpoint of segment GS. Let F be the angle bisector of angle GDS. Point R is the midpoint of segment KD. Point Y is the midpoint of segment SR. Let Q be the reflection of line F through point Y. Point O lies on line Q. Point A is the reflection of point O across line F. Find the distance between points A and O. Express your answer as a decimal number rounded to 2 decimal places.

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


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Let OF = 5. Point N is the reflection of point F across point O. Let C be the vector from point F to point O. Let Q be the perpendicular bisector of segment FO. Point Y is the translation of point N by vector C. Let L be the reflection of point Y across point O. Line J passes through point F parallel to line Q. Let H be the reflection of line Q across line J. Let T be the line through point O parallel to line H. Point P is on line T with |PN| = 6. Point M is the reflection of point N across segment LP. Point U is the reflection of point M across segment FP. Point Z is the reflection of point U across point P. Find the length of segment LZ. 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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IN = 12. Point E is obtained by rotating point I by angle 5pi/6 about point N. Let X be the midpoint of segment EN. Line A passes through point X perpendicular to segment IE. Point D is the midpoint of segment EX. Point O is obtained by rotating point D by angle 11pi/12 about point I. Let T be the midpoint of segment ON. Let Q be the angle bisector of angle OET. Point R is the reflection of point I across line Q. Point Y is the reflection of point R across line A. What is the length of segment YT? 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 WF = 7. Let X be the vector from point F to point W. Point E is the translation of point W by vector X. Point I is obtained by rotating point W by angle pi/4 about point F. Let D be the rotation of point W by angle 11pi/12 around point E. Line Y passes through points W and D. Point G is the reflection of point I across point W. Let O be the rotation of point G by angle 5pi/6 around point E. Point M is the midpoint of segment DE. Point B is the midpoint of segment MG. Line V is the perpendicular bisector of segment BW. Point R is obtained by rotating point O by angle 2pi/3 about point M. Point H is the midpoint of segment RW. Line Y and segment HB intersect at point A. Point U is the reflection of point A across line V. Let K be the reflection of point B across point U. What is the length of segment KA? Express your answer as a decimal number rounded to 2 decimal places.

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


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Let KM = 5. Let V be the reflection of point M across point K. Let I be the reflection of point K across point V. Let P be the vector from point M to point I. Line C is the angle bisector of angle KVI. Point J is obtained by rotating point K by angle 5pi/12 about point M. Let U be point J translated by vector P. Line T is the perpendicular bisector of segment MI. Let O be the reflection of line T across line C. Line S is the reflection of line O across line T. Let F be the midpoint of segment MI. Point Y is the reflection of point F across line S. Find the length of segment UY. 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 JE = 8. Point W is obtained by rotating point E by angle pi/3 about point J. Point N is the midpoint of segment EJ. Point V is the reflection of point W across point E. Point B is the reflection of point V across point W. Point G is the midpoint of segment NB. Point L is the reflection of point G across segment WJ. Point P is the midpoint of segment LE. Segments WJ and NB intersect at point T. Find the length of segment PT. Express your answer as a decimal number rounded to 2 decimal places.

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Let ML = 2. Point G is obtained by rotating point M by angle 2pi/3 about point L. Point I is the midpoint of segment LG. Point T is the reflection of point I across point M. Point D is the reflection of point T across point I. Point F is the midpoint of segment ML. Let S be the reflection of point D across point F. Line P is the perpendicular bisector of segment ML. Point O is the reflection of point S across segment ML. Let K be the line through point O parallel to segment ML. Let E be the angle bisector of angle OMD. Point B is the reflection of point G across line P. Line Z is the reflection of line E across line K. Let X be the midpoint of segment TO. Point Q is the reflection of point X across line Z. Let W be the midpoint of segment QB. Find 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}


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Let KS = 12. Let O be the rotation of point K by angle 5pi/6 around point S. Line W is the angle bisector of angle KSO. Point B is obtained by rotating point K by angle pi about point O. Let F be the line through point B parallel to line W. Let A be the reflection of point K across point B. Let I be the midpoint of segment SA. Point N is the reflection of point A across line W. Let E be the reflection of line F across line W. Let X be the reflection of line W across line E. Let P be the midpoint of segment BI. Point M is the reflection of point N across line X. Let T be the line through points A and P. Line T and segment KS intersect at point G. Point Y is the reflection of point G across point S. What is the length of segment MY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QG = 7. Point B is obtained by rotating point Q by angle 2pi/3 about point G. Line I is the perpendicular bisector of segment GQ. Let J be the reflection of line I through point G. Let C be the reflection of line J across line I. Line M passes through point Q parallel to segment BG. Point N is the midpoint of segment BG. Point R is the reflection of point N across line M. Point E is the reflection of point Q across line C. Point W is the reflection of point E across point B. Point S is the reflection of point G across line M. Let L be the reflection of point W across point R. Point T is the reflection of point L across segment BG. Let D be the midpoint of segment ST. Find the length of segment DT. 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 PN = 9. Point L is the reflection of point N across point P. Point W is obtained by rotating point L by angle 5pi/6 about point N. Point S is the midpoint of segment NW. Line J is the angle bisector of angle WPS. Point R is the midpoint of segment LW. Let O be the line through point W perpendicular to line J. Point Z is the reflection of point S across line O. Let B be the line through point Z perpendicular to segment PL. Point H is the reflection of point R across line B. 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}


[User]
Let BH = 11. Line P passes through points H and B. Point G is obtained by rotating point B by angle 5pi/6 about point H. Let M be the reflection of point H across point B. Point I is the reflection of point G across line P. Let F be the midpoint of segment MI. Let U be the reflection of point H across point F. Find the distance between points M and U. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EC = 8. Let Q be the perpendicular bisector of segment CE. Let A be the rotation of point E by angle 5pi/12 around point C. Line V passes through point E parallel to line Q. Let I be the reflection of line V across line Q. Let K be the rotation of point E by angle 11pi/12 around point C. Point P is obtained by rotating point K by angle 7pi/12 about point E. Point R is the intersection of segment KP and line I. Point B is obtained by rotating point A by angle pi/12 about point R. Point U is the reflection of point K across line V. Point J is the reflection of point E across segment KP. Point M is the reflection of point B across segment JU. Find the distance between points M and A. Express your answer as a decimal number rounded to 2 decimal places.

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


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RC = 2. Let B be the reflection of point C across point R. Line Z passes through points B and R. Point E is obtained by rotating point C by angle pi/4 about point R. Let U be the vector from point R to point B. Point A is the translation of point E by vector U. Let N be the line through point E parallel to line Z. Let Y be the reflection of line Z across line N. Let H be the rotation of point A by angle pi/6 around point R. Let W be the line through points B and E. Point O is the reflection of point H across line W. Point J is the reflection of point R across line Y. Let L be the midpoint of segment OJ. Let D be the midpoint of segment OL. What is the length of segment OD? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OG = 7. Let P be the midpoint of segment GO. Let X be the rotation of point O by angle 5pi/6 around point G. Point R is the midpoint of segment PX. Point M is obtained by rotating point G by angle pi/6 about point X. Let I be the midpoint of segment RX. Point T is the reflection of point O across point G. Point H is the reflection of point X across point T. Let B be the reflection of point M across point I. Point E is the reflection of point X across point H. Find 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 HK = 7. Let M be the rotation of point H by angle pi/2 around point K. Let D be the midpoint of segment KH. Let I be the perpendicular bisector of segment DH. Line N is the reflection of line I across point K. Point L is the reflection of point M across line N. Find the distance between points L and H. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VZ = 8. Point K is the reflection of point V across point Z. Point E is obtained by rotating point V by angle 2pi/3 about point K. Let R be the vector from point V to point K. Let B be point E translated by vector R. Let F be the rotation of point Z by angle pi/3 around point V. Let T be the reflection of point B across point V. Let W be the reflection of point V across point F. Point C is the midpoint of segment TE. Let U be the midpoint of segment EC. Let N be the midpoint of segment UW. What is the distance between points N and U? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RQ = 11. Let D be the rotation of point Q by angle 11pi/12 around point R. Line B is the angle bisector of angle QDR. Let Y be the midpoint of segment DQ. Point N is the reflection of point Y across line B. Point F is the midpoint of segment YN. What is the distance between points N and F? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
DW = 5. Let O be the rotation of point W by angle pi/2 around point D. Let I be the rotation of point D by angle pi/3 around point W. Let Q be the angle bisector of angle IWO. Point M is the midpoint of segment WO. Line P passes through point I perpendicular to line Q. Line S is the perpendicular bisector of segment OI. Let V be the reflection of line P across line S. Let G be the reflection of line V through point M. Point B is the reflection of point D across line G. Point X is the midpoint of segment WB. What is the length of segment OX? 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 SH = 2. Point Y is obtained by rotating point S by angle 5pi/12 about point H. Point R is obtained by rotating point S by angle 2pi/3 about point H. Let D be the rotation of point Y by angle 2pi/3 around point R. Point T is the reflection of point Y across point R. Let M be the reflection of point D across point T. Point L is the reflection of point Y across segment HS. Let G be the midpoint of segment LM. What is the distance between points G and D? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LE = 7. Let V be the midpoint of segment LE. Let S be the perpendicular bisector of segment VE. Line H passes through point E perpendicular to line S. Point J is obtained by rotating point L by angle pi/12 about point E. Line Z is the perpendicular bisector of segment EJ. Point O is the reflection of point V across line Z. Let X be the line through point O parallel to line H. Point K is the reflection of point O across point E. Point G is the reflection of point J across line X. Find the length of segment GK. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let IH = 1. Point E is obtained by rotating point H by angle 7pi/12 about point I. Let X be the reflection of point I across point E. Let Z be the line through points H and X. Line L passes through point H perpendicular to line Z. Point W is the reflection of point X across line L. Find the distance between points W 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 FQ = 7. Let X be the line through points Q and F. Point D is obtained by rotating point Q by angle 5pi/6 about point F. Point R is obtained by rotating point D by angle pi/12 about point F. Point W is obtained by rotating point F by angle pi/4 about point R. Let S be the reflection of line X through point D. Line B is the perpendicular bisector of segment WR. Line Z is the reflection of line B across line S. Let V be the line through point Q parallel to line B. Point G is the intersection of lines V and Z. What is the distance between points G 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 GU = 9. Let P be the midpoint of segment GU. Let X be the midpoint of segment PU. Point I is obtained by rotating point X by angle pi/6 about point P. Let E be the perpendicular bisector of segment XU. Point T is the reflection of point G across line E. Line S passes through point P parallel to line E. Line L passes through point X perpendicular to line S. Point O is the reflection of point X across point T. Line F passes through point T perpendicular to line L. Let M be the rotation of point U by angle pi/6 around point T. Let R be the line through point I parallel to line L. Point J lies on line F at distance 10 from point O. Point Y is the reflection of point U across point M. Line N passes through point Y perpendicular to line R. Let A be the reflection of line N through point J. Point W is the intersection of lines R and A. Point H is the midpoint of segment TX. Find the distance between points W 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 WY = 12. Let C be the reflection of point Y across point W. Let D be the midpoint of segment YW. Point R is obtained by rotating point C by angle 5pi/6 about point D. Point K is the reflection of point C across point D. Let O be the midpoint of segment YK. Let J be the midpoint of segment OR. Point N is the midpoint of segment WJ. Find the length of segment JN. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let KV = 8. Let I be the rotation of point K by angle pi/6 around point V. Let T be the rotation of point K by angle 7pi/12 around point I. Point C is the reflection of point I across point T. Point A is the reflection of point C across segment KV. Point G is the reflection of point T across segment KV. Find the distance between points G and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let QD = 1. Point Y is the midpoint of segment DQ. Point N is obtained by rotating point Q by angle 3pi/4 about point Y. Point O is the reflection of point Q across point D. Let V be the reflection of point O across point N. Point W is the reflection of point V across point Q. Point P is the reflection of point N across point W. Find the distance between points Y and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let UW = 11. Point X is the reflection of point W across point U. Point I is obtained by rotating point W by angle pi/12 about point X. Let K be the midpoint of segment IU. Point R is the reflection of point U across segment KX. Let E be the midpoint of segment RI. Let H be the midpoint of segment RE. Find the length of segment RH. Express your answer as a decimal number rounded to 2 decimal places.

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


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Let LX = 4. Point T is the midpoint of segment LX. Let U be the vector from point T to point X. Point P is obtained by rotating point L by angle 5pi/6 about point X. Point C is the translation of point X by vector U. Point Z is the midpoint of segment XP. Point J is the reflection of point Z across segment TX. Let F be the line through point J perpendicular to segment PT. Point V is the reflection of point C across line F. Find the distance between points Z and V. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let UM = 5. Let X be the midpoint of segment MU. Let C be the perpendicular bisector of segment UX. Point F is the reflection of point M across line C. Let D be the rotation of point M by angle 3pi/4 around point F. Line L is the perpendicular bisector of segment MU. Let I be the midpoint of segment DX. Point Q is the reflection of point I across segment MU. Line P is the reflection of line C across line L. Point R is the intersection of line P and segment MU. Find the length of segment RQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BE = 9. Let Y be the line through points E and B. Point N is obtained by rotating point B by angle pi/6 about point E. Let P be the perpendicular bisector of segment EB. Point O is the reflection of point E across point N. Point I is the reflection of point O across line P. Point G is the reflection of point I across line Y. Find the measure of angle IGB, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let HU = 7. Let O be the midpoint of segment UH. Let J be the vector from point O to point U. Line L is the perpendicular bisector of segment HO. Point V is on line L with |VH| = 11. Let Y be the reflection of point H across point U. Point T is the translation of point Y by vector J. Line S passes through point T perpendicular to segment HO. Point P is the midpoint of segment VT. Line W passes through points U and P. Line G is the reflection of line W across line S. Point D is the reflection of point V across line G. Find the distance between points H 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 PK = 3. Point F is the midpoint of segment KP. Let G be the rotation of point K by angle pi/4 around point F. Let W be the line through point P perpendicular to segment PK. Let Q be the line through point G parallel to segment PK. Point T is the reflection of point P across point G. Line X is the reflection of line W across point T. Point E is the reflection of point K across line X. Point J is the reflection of point E across line Q. What is the length of segment JE? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BU = 9. Let W be the midpoint of segment UB. Point T is the midpoint of segment UW. Let A be the reflection of point T across point W. Find the distance between points A 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 HE = 7. Point M is the midpoint of segment HE. Let V be the vector from point E to point H. Let R be the perpendicular bisector of segment HE. Point S lies on line R at distance 9 from point H. Let X be the line through points H and S. Point T is the translation of point M by vector V. Point N is on line R. Let C be point H translated by vector V. Line O passes through point N parallel to line X. Point W is the reflection of point C across line O. Point A is the reflection of point T across segment WC. Point D is the midpoint of segment ST. What is the length of segment DA? Express your answer as a decimal number rounded to 2 decimal places.

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


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CX = 10. Let E be the reflection of point X across point C. Point Z is obtained by rotating point C by angle pi/4 about point X. Point Y is the reflection of point E across segment CZ. Point M is the midpoint of segment YC. Point L is the midpoint of segment CM. Point R is the reflection of point Y across segment MZ. What is the distance between points R and L? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VS = 10. Line C passes through points V and S. Point X is the reflection of point V across point S. Let N be the reflection of point X across point V. Line U passes through point N perpendicular to line C. Point K is the reflection of point S across line U. Find the distance between points N and K. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AY = 7. Point D is obtained by rotating point A by angle 7pi/12 about point Y. Let T be the vector from point Y to point A. Let U be point D translated by vector T. Line F is the angle bisector of angle AYD. Point Q is obtained by rotating point A by angle pi/2 about point D. Line O is the angle bisector of angle DYU. Let C be the line through point Q parallel to line F. Let E be the reflection of line C across line O. Lines C and E intersect at point X. What is the distance between points X and U? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
TN = 6. Let A be the rotation of point T by angle pi/6 around point N. Let F be the reflection of point A across point N. Let Q be the reflection of point A across point F. Let R be the reflection of point N across point Q. Let S be the angle bisector of angle TRQ. Point J is the reflection of point N across line S. Find the length of segment AJ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let QZ = 9. Let R be the rotation of point Q by angle pi/4 around point Z. Let O be the rotation of point Q by angle 3pi/4 around point Z. Let S be the reflection of point R across point Q. Let D be the midpoint of segment SZ. Point J is the reflection of point D across segment OQ. Line W is the perpendicular bisector of segment JQ. Point T is the reflection of point Z across line W. Let M be the reflection of point D across point T. Find the length of segment MT. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EU = 10. Let X be the midpoint of segment EU. Line G passes through points E and X. Line V passes through point E perpendicular to line G. Let Y be the midpoint of segment XE. Point J is on line V with |JU| = 3. Point L is the reflection of point X across point U. Point K is the reflection of point J across point X. Point A is the midpoint of segment LK. Let P be the reflection of point A across point Y. What is the distance between points P 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 YL = 12. Let M be the vector from point Y to point L. Let E be point L translated by vector M. Point X is obtained by rotating point L by angle 11pi/12 about point Y. Let I be the reflection of point E across point Y. Let N be the angle bisector of angle IXY. Let T be the line through point Y parallel to line N. Point W is the reflection of point L across line T. Find the length of segment EW. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GO = 7. Let S be the rotation of point O by angle 7pi/12 around point G. Let W be the perpendicular bisector of segment GO. Line T is the reflection of line W across point S. Line I is the reflection of line T across line W. Point D is the reflection of point S across line I. Point B is the midpoint of segment GO. Let Q be the midpoint of segment DB. What is the distance between points S and Q? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
US = 9. Let R be the midpoint of segment SU. Point O is the midpoint of segment RS. Let Q be the rotation of point S by angle pi/2 around point R. Point K is the reflection of point O across segment QS. Let C be the line through points R and K. Point P is the intersection of segment QS and line C. Point V is the midpoint of segment PK. Find the distance between points V and O. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NZ = 1. Point S is obtained by rotating point Z by angle 7pi/12 about point N. Point K is the reflection of point N across segment ZS. Point L is the midpoint of segment SN. Point F is the reflection of point L across point S. Find the distance between points F and K. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
IT = 7. Let G be the rotation of point I by angle 7pi/12 around point T. Let P be the reflection of point T across point G. Point U is the reflection of point P across point I. Line K is the perpendicular bisector of segment IP. Let R be the line through point U parallel to line K. Point Z is the reflection of point G across line K. Let B be the reflection of line R across line K. Line Y passes through point Z perpendicular to line B. Line L passes through point G perpendicular to line Y. Let N be the reflection of point T across point Z. Point M is the reflection of point U across line L. Point V is the reflection of point Z across segment IG. Point X is the reflection of point M across point U. Let Q be the reflection of point N across point V. What is the distance between points X and Q? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let UT = 4. Point L is obtained by rotating point T by angle 5pi/12 about point U. Line J passes through points U and L. Point H is the reflection of point T across line J. Let O be the reflection of point U across point L. Point E is the reflection of point U across point O. What is the distance between points E and H? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
SW = 4. Line T is the perpendicular bisector of segment SW. Point F is the reflection of point S across point W. Point U lies on line T at distance 7 from point F. Point M is obtained by rotating point F by angle pi/3 about point W. Let L be the line through point M perpendicular to line T. Point Y is the reflection of point U across line L. Point V is obtained by rotating point M by angle pi/3 about point Y. Point O is the midpoint of segment SW. Point C is the reflection of point O across segment VU. Let N be the midpoint of segment VU. Find the length of segment CN. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
KD = 2. Let U be the perpendicular bisector of segment KD. Point F is the reflection of point D across point K. Point T is obtained by rotating point F by angle 11pi/12 about point D. Let E be the line through point D parallel to line U. Let Z be the rotation of point K by angle pi/12 around point T. Point C is obtained by rotating point K by angle pi/2 about point F. Line H is the reflection of line E across line U. Point W is the reflection of point C across line U. Point J is the reflection of point W across line H. Point M is the midpoint of segment ZJ. Let B be the reflection of point M across point K. Find the distance between points B and M. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let OB = 8. Point G is the midpoint of segment OB. M is constructed so that BM = 5. Let P be the midpoint of segment BM. Point Z is the midpoint of segment BP. Let X be the midpoint of segment PO. Point L is the reflection of point X across segment ZG. What is the length of segment ZL? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GJ = 3. Let N be the rotation of point G by angle 3pi/4 around point J. Let C be the reflection of point G across point J. Point O is obtained by rotating point G by angle 5pi/12 about point J. Let Q be the midpoint of segment CN. Point R is the reflection of point Q across segment JO. Find the distance between points R 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 RS = 12. Let F be the midpoint of segment SR. Point A is the midpoint of segment SF. Line Y is the perpendicular bisector of segment SA. Let O be the line through point A perpendicular to segment SF. Line X is the reflection of line Y across line O. Line X and segment SF intersect at point N. Find 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}