[User]
Let VY = 9. Point P is the reflection of point Y across point V. Point J is obtained by rotating point P by angle 11pi/12 about point Y. Let W be the perpendicular bisector of segment JP. Point X is obtained by rotating point V by angle 3pi/4 about point J. Let F be the line through points J and Y. Let C be the reflection of line W across line F. Let L be the reflection of line W through point V. Point B is the reflection of point Y across line L. Point Q is the reflection of point X across line C. Find the distance between points Q and B. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VC = 6. Point T is the midpoint of segment VC. Point E is the midpoint of segment TV. Point D is the reflection of point E across point C. Point R is the reflection of point C across point V. Let X be the midpoint of segment ET. Point B is the reflection of point D across point R. What is the distance between points B 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 ZS = 7. Let N be the vector from point Z to point S. Let D be the midpoint of segment ZS. Let F be the angle bisector of angle SDZ. Line T is the reflection of line F across point S. Line W is the reflection of line F across point Z. Point L lies on line T at distance 9 from point Z. Let M be the angle bisector of angle SLD. Let Y be the line through point S parallel to line M. Point B is the midpoint of segment ZL. Let K be point L translated by vector N. Let P be point B translated by vector N. Let Q be the line through point P perpendicular to line Y. Point H is the reflection of point B across line Q. Point U is the midpoint of segment ZH. Point A is the reflection of point K across point U. Point C is the reflection of point A across line W. What is the length of segment CZ? 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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KD = 10. Let N be the rotation of point K by angle 5pi/6 around point D. Let F be the rotation of point K by angle pi/4 around point D. Point X is the reflection of point N across point D. Point V is the midpoint of segment FX. Point E is the midpoint of segment XD. Line Q passes through point F perpendicular to segment FX. Let T be the midpoint of segment EV. Point O is the reflection of point V across line Q. Point S is the midpoint of segment FT. What is the length of segment SO? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IC = 2. Point P is the midpoint of segment CI. Line F passes through points P and C. Let Z be the line through point P perpendicular to line F. Point A is on line F with |AI| = 5. Let K be the vector from point I to point A. Line L passes through point I parallel to line Z. Point X is on line L with |XP| = 3. Point Y is the translation of point X by vector K. Point G is on line L with |GX| = 10. Line O passes through point G perpendicular to line Z. Let V be the line through points G and Y. Point W is the intersection of lines F and V. Let Q be the reflection of line O through point P. Point E is the reflection of point X across point W. Point B is the reflection of point E across line Q. Find the length of segment BE. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BQ = 8. Line K passes through points B and Q. Point A is the midpoint of segment BQ. Point N lies on line K at distance 9 from point B. Point X is on line K. Let V be the rotation of point N by angle pi/3 around point X. Let M be the midpoint of segment VB. Let T be the midpoint of segment AV. Point Y is the reflection of point M across segment AV. Find the distance between points Y and T. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JT = 9. Point D is obtained by rotating point T by angle pi/12 about point J. Let R be the vector from point J to point T. Let N be point T translated by vector R. Point H is the midpoint of segment JD. Point M is the midpoint of segment HD. Point C is the midpoint of segment HN. Let K be the reflection of point M across point C. What is the distance between points H and K? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
UK = 11. Let H be the midpoint of segment UK. Let A be the rotation of point H by angle pi/3 around point U. Let X be the vector from point U to point A. Point O is the reflection of point K across point U. Let D be point H translated by vector X. Let G be the line through points U and D. Point J is obtained by rotating point A by angle pi about point U. Line Q is the reflection of line G across point K. Point S is the reflection of point J across segment AO. Segments SH and AO intersect at point E. Point B is the reflection of point S across line Q. Find the distance between points B and E. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let TZ = 9. Point I is obtained by rotating point T by angle pi/6 about point Z. Let D be the midpoint of segment ZT. Let H be the rotation of point D by angle pi/4 around point Z. Let K be the midpoint of segment DI. Let R be the midpoint of segment TI. Line P is the perpendicular bisector of segment KD. Point F is the reflection of point H across line P. Let E be the reflection of point R across point F. Find the distance between points T and E. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GZ = 8. Let J be the midpoint of segment GZ. Let D be the angle bisector of angle JZG. W is placed 9 units away from Z. Let L be the angle bisector of angle GJW. Point K lies on line L at distance 4 from point J. Point P is the reflection of point K across line D. Point A is the reflection of point J across segment GW. Point H is the reflection of point P across segment GW. What is the distance between points H 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 PF = 3. Point K is the reflection of point P across point F. Point V is obtained by rotating point K by angle 7pi/12 about point P. Point E is obtained by rotating point K by angle 11pi/12 about point P. Let O be the perpendicular bisector of segment EV. Let N be the reflection of line O through point K. Let X be the midpoint of segment KE. Point C is the reflection of point X across line N. What is the distance between points C 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 RE = 6. Point I is the midpoint of segment ER. Let L be the midpoint of segment RI. Line J is the perpendicular bisector of segment LE. Let D be the midpoint of segment RL. Point C is the midpoint of segment LD. Let S be the reflection of point C across point D. Point O is the midpoint of segment RC. Let K be the rotation of point L by angle 5pi/6 around point C. Point M is the midpoint of segment KE. Point U is the reflection of point O across line J. Let N be the midpoint of segment SM. Point W is the reflection of point N across segment CM. Point A is the reflection of point U across segment CM. Find the distance between points A and W. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XK = 6. Let W be the rotation of point X by angle pi/2 around point K. Let A be the midpoint of segment XK. Point Z is the reflection of point A across point K. Point I is the reflection of point K across point W. Let S be the reflection of point K across point Z. Find the distance between points I and S. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WE = 6. Let S be the rotation of point E by angle 5pi/12 around point W. Line G passes through point S perpendicular to segment SW. Let C be the line through point E parallel to line G. Let P be the reflection of point W across point S. Point Q is the midpoint of segment SE. Point J is the reflection of point E across segment QP. Let B be the line through point W parallel to segment QP. Lines B and C intersect at point A. Let M be the midpoint of segment JP. What is 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}


[User]
Let AJ = 5. Let H be the rotation of point A by angle 5pi/12 around point J. Let Q be the reflection of point J across point H. Let I be the line through point H parallel to segment AQ. Let K be the perpendicular bisector of segment AQ. Point S is the reflection of point J across line I. Let R be the reflection of point S across point Q. Point V is the reflection of point R across line K. Point T is the midpoint of segment VQ. Let B be the midpoint of segment HQ. Find the length of segment TB. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let NB = 1. Point Y is the midpoint of segment NB. Let V be the reflection of point B across point N. Line Q is the perpendicular bisector of segment YV. Point I is the reflection of point B across line Q. Point F is the midpoint of segment VI. 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]
EG = 4. Let O be the midpoint of segment GE. Let Q be the rotation of point G by angle 5pi/12 around point E. Let B be the rotation of point E by angle pi around point Q. Let D be the midpoint of segment QE. Let S be the perpendicular bisector of segment GO. Point P lies on line S. Line M passes through point P parallel to segment QB. Let R be the line through point Q perpendicular to line M. Point X is the reflection of point G across point E. Point F is the reflection of point D across line R. Find the length of segment XF. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let IK = 8. Point J is obtained by rotating point K by angle pi/3 about point I. Point E is obtained by rotating point J by angle pi about point I. Line B is the perpendicular bisector of segment IE. Point G is on line B with |GJ| = 12. Point Q is the reflection of point G across segment IE. Point R is the midpoint of segment QK. Let C be the midpoint of segment JQ. Find the length of segment RC. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EA = 3. Let S be the perpendicular bisector of segment AE. Line I is the reflection of line S across point E. Point D lies on line I. Line Y is the reflection of line S across point D. Point H is the reflection of point D across line Y. 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]
RS = 2. Point E is the reflection of point R across point S. Point Q is the reflection of point S across point R. Let Y be the angle bisector of angle SRQ. Let H be the angle bisector of angle ESQ. Let Z be the reflection of line Y through point Q. Let X be the midpoint of segment QE. Line K passes through point Q parallel to line H. Point P is the reflection of point X across line Z. Point C lies on line K at distance 1 from point S. Point F is the reflection of point S across point E. Point D is the midpoint of segment CF. Point N is the midpoint of segment PD. Find the distance between points P and N. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BW = 10. Let T be the rotation of point W by angle pi/4 around point B. Point Q is the reflection of point W across point T. Line N passes through points Q and B. Line P passes through point T perpendicular to line N. Point R is the reflection of point Q across line P. Find the length of segment RT. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AK = 12. Point D is the midpoint of segment KA. Line W is the perpendicular bisector of segment KD. Point R is the reflection of point A across line W. Let S be the rotation of point D by angle 2pi/3 around point R. Let H be the angle bisector of angle RSA. Line O passes through point A perpendicular to line H. Point X is the reflection of point R across point S. Line G is the reflection of line O across point D. Point M is the intersection of line G and segment XA. What is the distance between points A 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 LK = 10. Point A is obtained by rotating point L by angle 2pi/3 about point K. Line B is the perpendicular bisector of segment LK. Let S be the midpoint of segment LA. Let M be the reflection of point S across point A. Point P is the reflection of point A across line B. Point R is the reflection of point P across segment LA. Let D be the midpoint of segment MP. Point Y is the reflection of point R across segment PD. What is the distance between points L and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NF = 11. Let W be the line through point N perpendicular to segment NF. Line H is the perpendicular bisector of segment NF. Point Y is on line W with |YF| = 1. Let V be the midpoint of segment NF. Line E is the reflection of line W across line H. Point D is the midpoint of segment VF. Line A passes through point D parallel to line E. Point Z is the reflection of point Y across line A. Find the distance between points Z and D. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NS = 6. Let G be the rotation of point N by angle pi/12 around point S. Let P be the midpoint of segment NS. Point D is the reflection of point G across segment NS. Let X be the rotation of point N by angle 3pi/4 around point D. Let R be the line through point X perpendicular to segment NS. Let J be the reflection of point X across point P. Point W is the reflection of point D across line R. Point H is the midpoint of segment GJ. What is the length of segment HW? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let NJ = 8. Point L is obtained by rotating point J by angle pi/2 about point N. Point Y is the reflection of point J across point N. Line M is the perpendicular bisector of segment YJ. Let T be the reflection of point N across point J. Point V lies on line M. Point W lies on line M at distance 12 from point T. Let K be the reflection of point W across point V. Point Z is the midpoint of segment YK. Line C passes through points L and Z. Point X is the reflection of point V across line C. Find the length of segment ZX. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IK = 1. B is placed 1 units away from K. Point U is obtained by rotating point B by angle 7pi/12 about point K. Let L be the rotation of point B by angle 3pi/4 around point K. Point M is obtained by rotating point U by angle 7pi/12 about point L. Point W is the midpoint of segment MK. Let Z be the line through points W and U. Point E is the reflection of point B across line Z. 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]
JR = 1. Line Z is the perpendicular bisector of segment JR. Let E be the vector from point J to point R. Point K is the midpoint of segment RJ. Let T be point R translated by vector E. Let I be the line through point J parallel to line Z. Let N be the perpendicular bisector of segment KT. Let F be the rotation of point T by angle 3pi/4 around point R. Let D be the line through point T parallel to line I. Let B be the line through point F parallel to line D. Point U is the reflection of point J across line B. Point L is the reflection of point U across line N. Point A is the reflection of point F across point L. Point X is the reflection of point J across point U. Find the length of segment XA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LV = 6. Let R be the reflection of point L across point V. Let N be the reflection of point V across point L. Let X be the rotation of point L by angle pi around point R. Line J is the perpendicular bisector of segment LN. Point H is the midpoint of segment VX. Let Z be the reflection of point L across point N. Point D lies on line J at distance 4 from point R. Line G passes through point R parallel to segment DZ. Let P be the line through point N perpendicular to segment NL. Line F passes through point Z perpendicular to segment NL. Line O is the reflection of line F across line P. Point U is the intersection of lines O and G. Find the distance between points H and U. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KM = 8. Point A is the midpoint of segment KM. Point C is the reflection of point A across point K. Point R is the reflection of point C across point A. Point O is obtained by rotating point A by angle pi/4 about point M. Let Z be the rotation of point C by angle 7pi/12 around point A. Let G be the rotation of point O by angle 11pi/12 around point A. Line D is the perpendicular bisector of segment KC. Point J is the midpoint of segment GZ. Point H is the reflection of point J across line D. Point W is the reflection of point H across segment KM. What is the distance between points R and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BM = 1. Point P is the midpoint of segment MB. Point F is obtained by rotating point B by angle 3pi/4 about point M. Let H be the rotation of point P by angle pi/12 around point M. Let K be the perpendicular bisector of segment FB. Point L is the reflection of point H across line K. Point Y is the reflection of point H across segment BL. What is the distance between points L and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GX = 10. Let O be the perpendicular bisector of segment GX. J is constructed so that XJ = 9. Let S be the midpoint of segment JX. Let K be the line through points S and G. Let P be the reflection of point G across point S. Line V is the reflection of line O across line K. Line Y passes through point P parallel to line V. Point W is the reflection of point J across line Y. What is the distance between points P and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NJ = 2. Let Q be the perpendicular bisector of segment JN. Line I is the reflection of line Q across point N. Point C is the reflection of point J across point N. Let D be the vector from point C to point J. Let R be point J translated by vector D. Let W be the reflection of line I across line Q. Let M be the reflection of line W across line I. Point K is the reflection of point N across line M. Let L be the midpoint of segment JR. Let H be the line through point L perpendicular to segment JC. Point T is the reflection of point K across line H. What is 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]
Let MS = 11. Point P is obtained by rotating point S by angle 5pi/6 about point M. Let V be the line through points P and M. Let I be the midpoint of segment SM. Let O be the midpoint of segment SI. Let N be the angle bisector of angle MPI. Line A is the reflection of line N across point I. Point C is the intersection of lines V and A. What is the distance between points C 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 WH = 1. Line T is the perpendicular bisector of segment HW. Point K is the midpoint of segment WH. Point I is obtained by rotating point W by angle pi/12 about point K. Line E passes through points I and H. Line C is the reflection of line E across point K. Point M is the reflection of point H across line C. Point D lies on line T. Line L passes through points D and M. Point P is the reflection of point W across line L. Find the length of segment PM. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PB = 6. Let N be the perpendicular bisector of segment BP. Point I lies on line N at distance 1 from point P. Point A lies on line N at distance 4 from point B. Let K be the midpoint of segment BA. Let L be the midpoint of segment AI. Line M passes through point L perpendicular to line N. Point S is the reflection of point K across line M. What is the distance between points S 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 VG = 2. Line C is the perpendicular bisector of segment VG. Let O be the midpoint of segment GV. Line X is the reflection of line C across point G. Point Z is obtained by rotating point V by angle 11pi/12 about point G. Let T be the reflection of line X across line C. Line P is the reflection of line C across line T. Point D is the reflection of point G across point V. Point Y lies on line P at distance 2 from point O. Let R be the perpendicular bisector of segment ZD. Point Q lies on line R. Point A is the midpoint of segment YQ. Let M be the midpoint of segment AQ. Point I is the reflection of point D across segment MO. Find the distance between points I and O. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BA = 3. Point D is obtained by rotating point B by angle 2pi/3 about point A. Let K be the perpendicular bisector of segment AB. Let E be the line through points B and A. Let V be the reflection of line K through point D. Lines E and V intersect at point Z. What is the distance between points B 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 ET = 6. Point C is obtained by rotating point E by angle 11pi/12 about point T. Line V is the perpendicular bisector of segment EC. Let S be the reflection of line V through point E. Let O be the reflection of point E across point T. Point G is the reflection of point O across line S. Point P is the reflection of point E across point C. Let F be the midpoint of segment TG. Find the length of segment FP. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let CV = 5. Let L be the midpoint of segment CV. Let J be the midpoint of segment VL. Let A be the rotation of point L by angle 3pi/4 around point V. Point T is the reflection of point C across point A. Point B is obtained by rotating point J by angle 5pi/6 about point C. Let E be the rotation of point T by angle pi/3 around point B. Let X be the reflection of point C across point B. Let H be the angle bisector of angle AXE. Let Q be the midpoint of segment XJ. Point Z is the reflection of point Q across line H. What is the distance between points Z and T? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let HI = 9. Let M be the reflection of point I across point H. Let C be the reflection of point I across point M. Point K is the reflection of point H across point C. Let V be the vector from point H to point I. Point N is obtained by rotating point M by angle 3pi/4 about point K. Let U be point N translated by vector V. Line T is the angle bisector of angle UNI. Let Q be the line through point C perpendicular to line T. Point O is the reflection of point C across line T. Point S is the reflection of point C across point O. Point R is the reflection of point I across line Q. Point X is the midpoint of segment OR. Find the length of segment XS. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JW = 10. Point H is the reflection of point W across point J. Let L be the reflection of point W across point H. Let S be the line through points L and J. Line M passes through point H perpendicular to line S. Let N be the rotation of point J by angle 3pi/4 around point L. Let Z be the reflection of line M through point W. Let F be the reflection of line M across line Z. Line B is the reflection of line M across line F. Point Y is the reflection of point N across line B. Point P lies on line F. Point K is the midpoint of segment YP. Point C is the reflection of point H across segment YK. Find the distance between points Y and C. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AC = 1. Let N be the reflection of point A across point C. Point I is the reflection of point N across point A. Point D is the midpoint of segment AI. Let W be the rotation of point A by angle 3pi/4 around point D. Let X be the rotation of point W by angle 7pi/12 around point N. Let L be the midpoint of segment DX. Point B is the midpoint of segment IX. Find the distance between points B and L. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IE = 4. Point A is obtained by rotating point I by angle 11pi/12 about point E. Let P be the midpoint of segment EA. Let K be the angle bisector of angle PAI. Line W passes through point I parallel to line K. Point B is the reflection of point P across line W. Let D be the midpoint of segment IP. Let N be the reflection of point D across point A. Point T is the reflection of point E across line W. Point J is the reflection of point T across segment IA. Let R be the reflection of point J across point N. Point S is the midpoint of segment BA. What is the distance between points S and R? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let EO = 3. Let H be the rotation of point O by angle 5pi/6 around point E. Let J be the rotation of point O by angle 3pi/4 around point H. Let F be the vector from point H to point O. Let S be point E translated by vector F. Let X be the rotation of point S by angle 5pi/6 around point J. Line Y is the angle bisector of angle HSE. Point N is the reflection of point X across line Y. Point M is the reflection of point J across line Y. Let B be the perpendicular bisector of segment JO. Point W is the midpoint of segment EO. Let Z be the reflection of point W across point N. Point U is the reflection of point Z across line B. What is the length of segment UM? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FS = 9. Point V is the midpoint of segment SF. Let C be the perpendicular bisector of segment SV. Line Y is the reflection of line C across point S. Point O is the reflection of point F across line Y. Find 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]
SN = 6. Let P be the vector from point S to point N. Let A be point N translated by vector P. Point Y is obtained by rotating point N by angle 5pi/12 about point A. Let W be the reflection of point N across point S. Point R is the midpoint of segment AY. Let O be the midpoint of segment AR. Line F is the perpendicular bisector of segment AY. Let V be the line through points W and R. Line Z is the reflection of line V across line F. Segment SN and line Z intersect at point K. What is the length of segment OK? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OK = 9. Point X is the reflection of point K across point O. Point Q is obtained by rotating point X by angle pi/6 about point O. Let M be the perpendicular bisector of segment XQ. Line S passes through point O perpendicular to line M. Point N is the midpoint of segment OK. Let I be the line through point N perpendicular to line S. Point F is on line I. Point R is the midpoint of segment FQ. Point P is the reflection of point X across point R. Find the distance between points F and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OT = 1. Point E is the midpoint of segment TO. Let H be the vector from point E to point O. Point A is the midpoint of segment OE. Line F is the perpendicular bisector of segment TO. Line C is the reflection of line F across point O. Point W is the midpoint of segment TE. Let I be the reflection of line F across line C. Let Q be point O translated by vector H. Let R be the rotation of point E by angle pi around point T. Point B is the reflection of point Q across point W. Line S is the perpendicular bisector of segment QO. Line J is the reflection of line S across line C. Line V passes through point B parallel to line I. Let G be the reflection of line J across line V. Point L is the reflection of point A across line G. 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]
XQ = 7. Point P is the reflection of point Q across point X. Let R be the rotation of point X by angle 5pi/12 around point P. Point U is the midpoint of segment QR. Line B is the perpendicular bisector of segment QX. Line F is the reflection of line B across point X. Let V be the perpendicular bisector of segment UQ. Let I be the reflection of line V across line B. Point M is the reflection of point U across segment QX. Point Z is the intersection of lines F and I. What is the length of segment ZM? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
MW = 4. Let N be the rotation of point W by angle pi/6 around point M. Let U be the line through points M and N. Let F be the rotation of point M by angle pi/3 around point W. Line D passes through points M and W. Line A is the reflection of line U across line D. Let O be the reflection of point F across point M. Point P is obtained by rotating point F by angle pi/12 about point O. Line J passes through points F and P. Point T is the intersection of lines J and A. Line S is the reflection of line U across point T. Lines J and S intersect at point R. Find the distance between points F and R. Express your answer as a decimal number rounded to 2 decimal places.

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


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Let OM = 9. Let X be the reflection of point M across point O. Let F be the line through points M and X. Point N is on line F. Let J be the rotation of point O by angle 7pi/12 around point X. Let Q be the reflection of point M across point N. Point W is the reflection of point Q across point J. Point Z is the midpoint of segment WJ. Line K passes through point Z perpendicular to line F. Point R is the reflection of point O across line K. What is the distance between points R and W? Express your answer as a decimal number rounded to 2 decimal places.

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


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YU = 6. Let S be the line through points U and Y. Let X be the rotation of point U by angle pi/4 around point Y. Line W is the perpendicular bisector of segment YX. Line T is the reflection of line W across point Y. Let D be the line through point U perpendicular to line S. Line I is the reflection of line T across line S. Let V be the line through point X perpendicular to line T. Let E be the midpoint of segment XY. Let N be the midpoint of segment EU. Line B passes through point N parallel to line D. Line L passes through point N parallel to line S. Point Z is the reflection of point N across line I. Let H be the reflection of line L across line V. Let F be the rotation of point Y by angle pi/2 around point Z. Let G be the rotation of point F by angle 7pi/12 around point U. Segment FG and line H intersect at point R. Let M be the reflection of point E across point R. Point K is the intersection of lines T and B. Point J is the reflection of point K across segment GX. Point P is the reflection of point M across segment FG. Find the length of segment JP. Express your answer as a decimal number rounded to 2 decimal places.

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Let GQ = 1. Let O be the vector from point Q to point G. Let J be the line through points G and Q. Let R be the midpoint of segment QG. Let S be point R translated by vector O. Let L be the perpendicular bisector of segment QG. Point Z is obtained by rotating point R by angle 2pi/3 about point G. Let U be the reflection of point G across point Z. Let F be the reflection of point Q across point U. Point Y is obtained by rotating point Z by angle 5pi/6 about point U. Let K be the midpoint of segment YS. Point D is the reflection of point Z across line L. Point W is the midpoint of segment UQ. Line E passes through point F parallel to line J. Line M passes through points D and W. Line M and segment GK intersect at point C. Point V is the reflection of point K across line M. Let X be the reflection of point G across point C. Let P be the line through point X perpendicular to line E. Point T is the reflection of point V across line P. What is the length of segment ZT? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ZP = 5. Let C be the midpoint of segment ZP. Line F passes through point P perpendicular to segment ZP. Point M is obtained by rotating point C by angle pi/6 about point P. Point N is the reflection of point C across point Z. Point L is the reflection of point Z across line F. Line Q passes through point N parallel to line F. Line A is the reflection of line Q across line F. Point D is the reflection of point L across line A. Find the length of segment DM. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let TY = 1. Line A passes through points Y and T. Point V is the reflection of point T across point Y. Line U passes through point T perpendicular to line A. Point K is the reflection of point Y across line U. Find the distance between points V and K. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let KH = 9. Point A is obtained by rotating point H by angle 11pi/12 about point K. Point L is the reflection of point K across point A. Point S is obtained by rotating point L by angle pi/12 about point K. Let M be the rotation of point K by angle pi/6 around point S. Let V be the reflection of point A across point M. Let Z be the reflection of point V across point L. Line J passes through point Z perpendicular to segment KS. Point W is the reflection of point Z across segment KS. Point N is the reflection of point A across line J. Let E be the midpoint of segment NW. Find the distance between points K and E. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
IU = 12. Let T be the midpoint of segment UI. Line Z passes through points U and T. Let F be the rotation of point U by angle 2pi/3 around point I. Let P be the midpoint of segment IT. Point E is on line Z with |EP| = 2. Point Y is the reflection of point T across point I. Line O is the perpendicular bisector of segment YE. Line X passes through point P parallel to line O. Line K is the reflection of line X across point Y. Point J is the intersection of lines Z and K. What is the length of segment FJ? Express your answer as a decimal number rounded to 2 decimal places.

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Let LO = 6. Let S be the reflection of point L across point O. Point R is the midpoint of segment OL. Line X is the perpendicular bisector of segment SO. Point A is on line X with |AO| = 9. Line M passes through points A and R. Line U passes through point R perpendicular to line M. Let B be the midpoint of segment OR. Point E is the reflection of point L across line U. Point V is the reflection of point B across line U. What is the length of segment VE? 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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OV = 10. Let G be the reflection of point V across point O. Point T is obtained by rotating point O by angle 2pi/3 about point G. Line U is the angle bisector of angle GTV. Point Q is the reflection of point T across point G. Line X is the reflection of line U across point Q. Point W is the reflection of point V across point Q. Point Z is the reflection of point V across line X. Let A be the reflection of point W across point O. Point D is the reflection of point A across line U. What is the length of segment ZD? Express your answer as a decimal number rounded to 2 decimal places.

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Let XP = 6. Let Y be the perpendicular bisector of segment XP. Point W is obtained by rotating point P by angle 7pi/12 about point X. Let G be the reflection of line Y through point P. Let Z be the reflection of point X across point W. Point E is the midpoint of segment PW. Point R is the reflection of point Z across line G. Let S be the angle bisector of angle PZR. Point O is the reflection of point E across line S. Find the length of segment OZ. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let CF = 5. Point U is the reflection of point C across point F. Let J be the perpendicular bisector of segment CF. Point W is the reflection of point U across point C. Point X is on line J with |XU| = 12. Let Y be the line through points C and X. Point G is the reflection of point U across line Y. Point T is the reflection of point U across segment XW. 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]
Let HO = 3. Point K is obtained by rotating point H by angle pi/12 about point O. Point Z is the reflection of point H across segment OK. Let F be the rotation of point O by angle pi/12 around point K. Let L be the angle bisector of angle ZKO. Let Y be the midpoint of segment ZH. Let P be the reflection of line L through point Z. Let R be the line through points H and Y. Lines R and P intersect at point J. Point N is the midpoint of segment ZF. What is the distance between points N and J? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let BL = 8. Point I is the reflection of point B across point L. Let X be the angle bisector of angle ILB. Line W is the perpendicular bisector of segment LB. Point F is the midpoint of segment IL. Point H is obtained by rotating point B by angle 2pi/3 about point L. Let Q be the line through point I parallel to line W. Line N is the reflection of line X across line Q. Point D is the reflection of point F across point I. Point O is the reflection of point L across line N. Point P is obtained by rotating point D by angle pi about point L. Point T is the midpoint of segment OL. Point V is the reflection of point D across point H. Let G be the midpoint of segment TV. Point E is the reflection of point V across line Q. Point Y is the reflection of point E across point I. Point C is the reflection of point Y across point G. Find the distance between points P and C. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EB = 10. Let Z be the vector from point E to point B. Let F be the vector from point B to point E. Line T passes through points B and E. Let V be the vector from point E to point B. Point U is the reflection of point E across point B. Let Y be the rotation of point U by angle 5pi/12 around point B. Point G is the reflection of point Y across line T. Let K be the line through points E and Y. Line A is the reflection of line K across line T. Point W is the translation of point U by vector Z. Let R be the reflection of point Y across point W. Point J is the midpoint of segment GR. Let M be point R translated by vector F. Let N be point J translated by vector V. Let P be point R translated by vector Z. Point Q is the reflection of point P across line A. Let H be the midpoint of segment UW. Let I be the midpoint of segment QN. Let C be the reflection of point M across point I. 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]
RZ = 7. Let K be the reflection of point R across point Z. Point J is the reflection of point Z across point R. Point W is the midpoint of segment JK. Let A be the rotation of point J by angle pi/12 around point Z. Point D is the midpoint of segment RJ. Point G is obtained by rotating point D by angle pi/3 about point W. Line T is the perpendicular bisector of segment GW. Point B is the midpoint of segment DK. Point L is the reflection of point B across point A. Point O is the reflection of point L across line T. What is the distance between points O and L? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RE = 12. Let V be the vector from point E to point R. Let J be the perpendicular bisector of segment RE. Point F is the translation of point R by vector V. Point A is the reflection of point E across point F. Point H is the reflection of point A across point E. Point W is on line J. Line K passes through point W perpendicular to line J. Let B be the line through point F perpendicular to line K. Point T lies on line B. Point U is the reflection of point T across segment AF. Point I is the reflection of point U across point H. What is the distance between points I and T? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let QR = 8. Let Y be the rotation of point Q by angle 5pi/12 around point R. Let B be the rotation of point R by angle 3pi/4 around point Y. Line M passes through points B and Y. Point O is the reflection of point R across point Q. Point J is obtained by rotating point R by angle pi/3 about point B. Let F be the reflection of line M through point J. Let N be the line through point R parallel to line F. Let Z be the reflection of point Q across point B. Line C is the reflection of line N across point B. Segment RZ and line C intersect at point H. What is the length of segment OH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let TJ = 7. Point L is obtained by rotating point J by angle 5pi/12 about point T. Point K is the reflection of point J across point T. Line U passes through point K perpendicular to segment LT. Line N is the reflection of line U across point J. Let D be the reflection of point T across point L. Point Y is obtained by rotating point K by angle 11pi/12 about point T. Line A passes through point T parallel to line N. Line M is the angle bisector of angle DTJ. Point V is obtained by rotating point L by angle pi/2 about point Y. Point W is the reflection of point K across point D. Point P is the midpoint of segment TL. Let F be the reflection of point V across point W. Let X be the line through point F parallel to line A. Point R is the reflection of point P across line M. Point O is the reflection of point J across line X. What is the length of segment RO? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let JP = 7. Let V be the rotation of point J by angle pi/6 around point P. Point A is the midpoint of segment PV. Line N is the angle bisector of angle VAJ. Let H be the line through point V parallel to line N. Point L is the reflection of point A across line H. Point O is the reflection of point P across point L. Find the length of segment OJ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ZN = 10. Point S is the reflection of point N across point Z. Point C is the midpoint of segment ZS. Line Y is the perpendicular bisector of segment CN. Point X is obtained by rotating point C by angle pi/2 about point N. Point M is obtained by rotating point S by angle 3pi/4 about point Z. Line A is the reflection of line Y across point Z. Point H is the reflection of point X across point C. Point L is the reflection of point M across line A. Point F is the reflection of point X across point H. Point K is the midpoint of segment HF. What is the distance between points L 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 = 12. Let Q be the perpendicular bisector of segment DT. Let K be the reflection of point T across point D. Let F be the rotation of point T by angle pi around point K. Let R be the midpoint of segment DF. Line S is the reflection of line Q across point T. Line M passes through points R and D. Let J be the rotation of point F by angle 7pi/12 around point R. Line O is the perpendicular bisector of segment DR. Let L be the midpoint of segment JD. Point A lies on line O. Let C be the reflection of line Q through point A. Lines C and M intersect at point Y. Point Z is the reflection of point L across line S. What is the length of segment YZ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OD = 11. Point H is obtained by rotating point O by angle pi/3 about point D. Let V be the rotation of point H by angle 3pi/4 around point O. Let G be the reflection of point V across point H. Let J be the reflection of point D across point O. Let E be the rotation of point J by angle 5pi/12 around point V. Let Y be the perpendicular bisector of segment OD. Line Z passes through point V perpendicular to line Y. Point T is the reflection of point G across line Z. Point Q is the reflection of point V across line Y. Line A passes through point O parallel to segment EQ. Point N is the reflection of point T across line A. Find the distance between points N 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 KI = 1. Let V be the rotation of point I by angle pi/2 around point K. Point U is obtained by rotating point V by angle 11pi/12 about point I. Let P be the midpoint of segment KI. Let C be the rotation of point U by angle pi/2 around point P. Point M is the reflection of point U across segment KV. Let O be the reflection of point U across point M. Let Y be the midpoint of segment KV. Let H be the reflection of point Y across point C. Let D be the angle bisector of angle HCV. Line A is the reflection of line D across point U. Point R is the reflection of point Y across line A. Find the distance between points R and O. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
VT = 12. Point O is the reflection of point T across point V. Let S be the midpoint of segment VO. Point G is obtained by rotating point O by angle 2pi/3 about point S. Let Z be the rotation of point V by angle pi/12 around point S. Let Q be the perpendicular bisector of segment OT. Line H is the angle bisector of angle GZO. Point W is the reflection of point T across line H. Let Y be the reflection of line Q across line H. Point R is the reflection of point O across point W. Point F is the reflection of point R across line Y. Let X be the line through point F perpendicular to segment VZ. Point J is the reflection of point V across line X. 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]
TP = 9. Point W is obtained by rotating point P by angle pi/12 about point T. Let F be the midpoint of segment TP. Let V be the angle bisector of angle TWP. Point Z is the reflection of point P across line V. Point G is the reflection of point F across line V. Let U be the midpoint of segment ZG. What is the length of segment UG? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
HZ = 10. Let N be the reflection of point H across point Z. Point I is obtained by rotating point H by angle pi/12 about point Z. Point P is obtained by rotating point N by angle pi/12 about point H. Let E be the rotation of point P by angle 11pi/12 around point N. Let C be the rotation of point E by angle 7pi/12 around point P. Line T is the perpendicular bisector of segment IC. Point U is the reflection of point P across line T. Point J is the reflection of point U across segment IC. What is the distance between points J and P? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let HJ = 7. Let C be the midpoint of segment HJ. Let V be the reflection of point C across point J. Line L passes through point C perpendicular to segment HJ. Let W be the rotation of point V by angle pi/12 around point H. Point I is the reflection of point H across point J. Point S is the reflection of point W across segment HJ. Point U is the reflection of point I across line L. Let D be the midpoint of segment US. Point B is obtained by rotating point I by angle pi/6 about point V. Let T be the rotation of point U by angle 3pi/4 around point J. Point M is the midpoint of segment BU. Let X be the angle bisector of angle CIT. Point G is the reflection of point M across segment HJ. Point R is the reflection of point G across line X. Let Z be the reflection of point U across point R. What is the distance between points D and Z? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KR = 6. Let J be the vector from point K to point R. Let U be the rotation of point R by angle 11pi/12 around point K. Point L is obtained by rotating point U by angle pi/4 about point K. Let X be point R translated by vector J. Line Y is the perpendicular bisector of segment LR. Line C passes through point U parallel to line Y. Point B is the reflection of point X across line C. Let E be the midpoint of segment LR. Let W be the reflection of point U across point E. What is the length of segment WB? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
PX = 4. Let I be the line through points X and P. Line L passes through point X perpendicular to line I. Point G is on line L with |GX| = 2. Point Q is the reflection of point X across point G. Line A is the perpendicular bisector of segment GP. Line Z passes through point G perpendicular to line A. Point B is the reflection of point X across segment PQ. Point U is the reflection of point X across line Z. Find the distance between points B and U. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OT = 10. Line E passes through points O and T. Line D is the perpendicular bisector of segment TO. Let A be the reflection of point O across point T. Line B is the reflection of line D across point O. Point F is obtained by rotating point T by angle 5pi/12 about point O. Let X be the reflection of line E through point F. Let Q be the line through point A perpendicular to segment FT. Point I is the reflection of point T across line Q. Point M is the midpoint of segment FT. Point H is obtained by rotating point M by angle 5pi/12 about point I. Let U be the reflection of line X through point H. Point Y is the intersection of lines U and B. Point W is the reflection of point Y across segment AM. Point P is the reflection of point W across segment FT. 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}


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EX = 1. Point G is the reflection of point X across point E. Line H is the perpendicular bisector of segment GX. Point C lies on line H at distance 4 from point G. J is constructed so that CJ = 1. Let D be the angle bisector of angle JXC. Point Q lies on line D at distance 5 from point J. Point R is the reflection of point Q across point X. Let O be the midpoint of segment CR. Point B is the midpoint of segment GR. Find the length of segment BO. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
CR = 8. Let Y be the line through points C and R. Point B is the reflection of point R across point C. Point U is obtained by rotating point C by angle pi about point R. Point N is obtained by rotating point B by angle pi/6 about point U. Let S be the midpoint of segment BC. Let X be the reflection of point N across point B. Let O be the angle bisector of angle UCN. Let W be the line through point B perpendicular to line O. Line A passes through point X parallel to line W. Point L is the intersection of lines Y and A. Find the distance between points L 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 GA = 11. Line U passes through points A and G. Point M is obtained by rotating point A by angle pi/4 about point G. Let H be the midpoint of segment AM. Let B be the line through points M and G. Point Y is the reflection of point H across line U. Point C is the reflection of point H across line B. What is the measure of angle CYH, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let FT = 1. Line X is the perpendicular bisector of segment TF. Let Y be the reflection of point F across point T. Line D is the reflection of line X across point T. Line M is the perpendicular bisector of segment FY. Let N be the reflection of point Y across point F. Point C is the reflection of point Y across point N. Let G be the reflection of point T across point F. Let H be the angle bisector of angle TFG. Let I be the reflection of line M across line D. Point Q is the reflection of point C across line I. Point U is the reflection of point Q across line H. Find the distance between points U and G. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let KO = 5. Let Z be the line through point O perpendicular to segment KO. Point R lies on line Z at distance 1 from point O. Point L is the reflection of point O across point K. Let A be the midpoint of segment KO. Point I is the reflection of point A across point L. Let G be the midpoint of segment KR. Point W is the reflection of point I across segment KR. Line S passes through points W and I. Point U is the reflection of point G across line S. Let Q be the midpoint of segment UI. Find the length of segment KQ. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ZY = 8. Let J be the midpoint of segment YZ. Let O be the reflection of point Y across point Z. Let H be the rotation of point O by angle pi/3 around point Z. Let R be the reflection of point J across point Z. Line G is the perpendicular bisector of segment HZ. Line P passes through point R parallel to line G. Point W is the reflection of point H across line P. What is the length of segment HW? Express your answer as a decimal number rounded to 2 decimal places.

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