[User]
FN = 6. Point L is obtained by rotating point N by angle 11pi/12 about point F. Line R is the angle bisector of angle LNF. Point S is obtained by rotating point N by angle 5pi/6 about point F. Let E be the rotation of point S by angle pi/2 around point N. Let A be the midpoint of segment FE. Let T be the line through points E and L. Point I is the reflection of point A across point N. Line P passes through point N parallel to line T. Point W is the reflection of point N across point A. Point K is the reflection of point I across line R. Line H passes through point W perpendicular to line P. Point X is the reflection of point I across line H. Find the length of segment XK. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
UA = 4. Let L be the vector from point U to point A. Point R is the reflection of point U across point A. Line T is the perpendicular bisector of segment RA. Point H is the midpoint of segment AU. Line W is the reflection of line T across point A. Line S is the perpendicular bisector of segment UH. Point N is the midpoint of segment HA. Let I be point H translated by vector L. Point C is the reflection of point N across point I. Let G be the reflection of line W through point N. Let E be the rotation of point H by angle 2pi/3 around point I. Point X is the reflection of point E across line G. Let M be the reflection of point X across point E. Let Y be the rotation of point E by angle 5pi/12 around point M. Let V be the angle bisector of angle YRU. Point K is the reflection of point X across point U. Line Q passes through point I parallel to line V. Point J is the reflection of point C across line V. Let B be the reflection of point M across point K. Point O is the reflection of point J across line Q. Point Z is the reflection of point O across line S. Find the distance between points Z and B. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PM = 3. Let Q be the rotation of point P by angle pi/2 around point M. Point Z is the reflection of point Q across point P. Let O be the midpoint of segment ZM. Let T be the reflection of point P across point Z. Point U is the midpoint of segment MT. Find the length of segment UO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let EI = 1. Let K be the midpoint of segment EI. Point M is obtained by rotating point K by angle 5pi/6 about point I. Point L is the reflection of point I across point M. Let Q be the reflection of point M across point K. Let G be the vector from point I to point Q. Point N is the translation of point K by vector G. Let Z be the line through points N and L. Point U is the reflection of point K across line Z. Let F be the reflection of point N across point L. Point C is the midpoint of segment FQ. What is the length of segment UC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GS = 7. Point D is obtained by rotating point G by angle 5pi/6 about point S. Point O is the reflection of point D across point G. Point V is obtained by rotating point S by angle 11pi/12 about point O. Let J be the angle bisector of angle DOV. Point P is the midpoint of segment GO. Point W is the reflection of point G across line J. Line E is the angle bisector of angle OPG. Point Y is the reflection of point W across line E. Point U is the reflection of point Y across segment PG. What is the distance between points U and O? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let HL = 8. Let E be the midpoint of segment LH. Let T be the reflection of point E across point L. Point N is the reflection of point H across point T. Let B be the line through point E perpendicular to segment LH. Let Y be the midpoint of segment NE. Line S is the reflection of line B across point H. Point W is the reflection of point H across line S. What is the distance between points Y and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ML = 12. Let X be the rotation of point M by angle 5pi/6 around point L. Point V is obtained by rotating point X by angle 3pi/4 about point M. Point Z is the reflection of point M across point L. Line B is the angle bisector of angle LMX. Line W is the reflection of line B across point X. Let Y be the line through point L parallel to line W. Let J be the midpoint of segment ZL. Point G is the intersection of segment VM and line Y. Find the distance between points J and G. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GH = 3. Line X is the perpendicular bisector of segment HG. Let U be the reflection of point H across point G. Point O is the midpoint of segment UG. Point E is the reflection of point H across point U. Let S be the rotation of point O by angle 5pi/6 around point U. Let P be the perpendicular bisector of segment EH. Let K be the reflection of point S across point U. Point B is the midpoint of segment HG. Let V be the reflection of point B across point K. Let W be the line through point V perpendicular to segment HG. Line A is the reflection of line P across line W. Let I be the reflection of line W across line A. Point R is the reflection of point S across line I. Point C is the reflection of point O across line X. What is the distance between points C and R? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OD = 6. Let U be the rotation of point O by angle pi/6 around point D. Let G be the midpoint of segment DO. Point Z is the reflection of point G across point U. Let N be the midpoint of segment OZ. Line X is the angle bisector of angle DZG. Let Y be the rotation of point Z by angle 7pi/12 around point N. Let C be the line through point G parallel to line X. Point S is the reflection of point Y across point U. Let K be the midpoint of segment OY. Point P is the reflection of point K across line C. Let R be the reflection of point S across point O. Point L is the reflection of point N across point P. Find the length of segment RL. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GX = 1. Point Q is the midpoint of segment GX. Let B be the reflection of point G across point X. Line V passes through point B perpendicular to segment GX. Point Z is the reflection of point G across line V. Let M be the reflection of point G across point Z. Point K is obtained by rotating point M by angle pi/12 about point G. Point A is the midpoint of segment QK. Let F be the reflection of point Q across point G. Point D is the reflection of point F across segment AK. Find 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 XN = 1. Let R be the reflection of point N across point X. Let I be the perpendicular bisector of segment NR. Point C is on line I with |CX| = 5. Let F be the midpoint of segment XR. Line V is the angle bisector of angle NRC. Let A be the reflection of point N across point F. Let S be the reflection of point A across point C. Let Q be the reflection of point X across point A. Point U is the reflection of point Q across line V. What is the length of segment US? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HN = 9. Line E is the perpendicular bisector of segment NH. Let V be the line through point H parallel to line E. Let S be the midpoint of segment NH. Point P lies on line V at distance 12 from point N. Point Z is the midpoint of segment NP. Let A be the reflection of point Z across point S. Point J is the reflection of point S across point A. Let Q be the midpoint of segment NA. What is the length of segment QJ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QE = 3. Let T be the rotation of point Q by angle pi/3 around point E. Let A be the vector from point T to point Q. Let F be the midpoint of segment TQ. Let D be the rotation of point F by angle 3pi/4 around point E. Let O be point D translated by vector A. Let V be the midpoint of segment TO. Line L is the angle bisector of angle TDQ. Line I is the angle bisector of angle QEF. Line N is the reflection of line I across line L. Point K is the reflection of point D across segment VE. Let W be the line through point F perpendicular to line I. Point P is the reflection of point K across line W. Point B is the reflection of point P across line N. What is the length of segment KB? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let QT = 12. Point Y is the reflection of point Q across point T. Line X passes through points Y and Q. Let M be the angle bisector of angle YTQ. Let F be the line through point Q perpendicular to line X. Point J is the reflection of point T across line F. Point V is the reflection of point J across line M. Find the distance between points V and J. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
UV = 2. Let Y be the rotation of point V by angle pi/3 around point U. Point X is the midpoint of segment YU. Point G is the midpoint of segment VX. Let W be the reflection of point Y across point G. Point P is the reflection of point W across segment UV. Let O be the midpoint of segment GW. Let F be the midpoint of segment WO. What is the distance between points P and F? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WQ = 5. Let T be the rotation of point W by angle pi/4 around point Q. Let H be the midpoint of segment QW. Point X is the midpoint of segment WH. Point F is the reflection of point T across point H. Point C is the midpoint of segment FX. What is the length of segment CT? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RD = 9. Point J is obtained by rotating point R by angle 7pi/12 about point D. Point B is the midpoint of segment RJ. Let T be the rotation of point R by angle 7pi/12 around point B. Line M passes through points D and T. Point H is the reflection of point B across line M. Let V be the line through point B parallel to segment JT. Point P is the reflection of point R across segment JT. Line K passes through points P and H. Lines K and V intersect at point F. What is the distance between points F 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 UT = 2. Point J is the midpoint of segment UT. Point A is the reflection of point U across point T. Let R be the rotation of point U by angle pi/3 around point J. Let S be the rotation of point J by angle 2pi/3 around point A. Let Y be the perpendicular bisector of segment UT. Line M is the angle bisector of angle ASJ. Let L be the line through point R parallel to segment UT. Let E be the reflection of line M across line Y. Lines E and L intersect at point O. Point X is the reflection of point U across segment SR. Find 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]
Let UN = 9. Point D is obtained by rotating point U by angle 2pi/3 about point N. Let E be the midpoint of segment NU. Line C is the perpendicular bisector of segment NU. Point M lies on segment NU. Let H be the reflection of point D across point E. Let P be the rotation of point U by angle pi/4 around point H. Let J be the reflection of line C through point U. Line I is the reflection of line J across line C. Point G is the reflection of point P across line I. Point Q is obtained by rotating point M by angle pi about point H. Point B is the reflection of point E across point P. Point O is the reflection of point Q across segment NU. Point Y is the reflection of point G across segment OQ. Point V is the reflection of point B across segment OQ. Find the distance between points V and Y. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YE = 6. Point H is the reflection of point Y across point E. Let O be the midpoint of segment YE. Let Q be the reflection of point H across point O. Point G is the midpoint of segment QO. What is the distance between points G and Q? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
TL = 3. Let Q be the midpoint of segment LT. Let Z be the perpendicular bisector of segment TQ. Point E is on line Z with |ET| = 2. Let R be the reflection of point Q across point L. Point C is the reflection of point R across point Q. Let G be the midpoint of segment RE. Point M is the reflection of point C across point G. What is the distance between points M and C? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LQ = 3. Let I be the vector from point Q to point L. Point X is the midpoint of segment QL. Point W is the translation of point X by vector I. Let O be the rotation of point X by angle pi/3 around point L. Let Y be the midpoint of segment OQ. Let B be the perpendicular bisector of segment OY. Line H passes through point O perpendicular to line B. Let F be the line through point X parallel to line H. Point P is the reflection of point W across line F. Point A is the reflection of point P across segment YW. Find the length of segment OA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
US = 3. Point Q is the reflection of point U across point S. Let P be the rotation of point U by angle 11pi/12 around point S. Let W be the vector from point P to point S. Line A passes through points U and S. Let L be the vector from point S to point Q. Point J lies on line A. Let Z be point S translated by vector W. Point Y is the translation of point P by vector L. Let N be the rotation of point Z by angle 11pi/12 around point S. Line E passes through point Q perpendicular to line A. Line M is the perpendicular bisector of segment NP. Let T be the reflection of point Y across point J. Point F is the reflection of point Z across line M. Let O be the line through point T perpendicular to line E. Point C is the reflection of point F across line O. What is the distance between points C and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WF = 3. Point X is obtained by rotating point W by angle 7pi/12 about point F. Point R is the midpoint of segment XF. Let N be the vector from point X to point R. Let B be the angle bisector of angle WFX. Point Y is the reflection of point X across point W. Line A passes through points Y and R. Let I be the reflection of line A across line B. Let Q be the rotation of point R by angle 5pi/12 around point Y. Let S be the midpoint of segment FR. Point D is the translation of point Y by vector N. Point P is the reflection of point Q across line I. Point T is the reflection of point D across line I. Let O be the line through points F and Q. Let H be the reflection of line O through point S. Segment PX and line H intersect at point E. 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 KH = 1. Let V be the midpoint of segment KH. Point Y is the reflection of point H across point K. Let R be the rotation of point Y by angle pi/3 around point V. Point S is the reflection of point V across point H. Line W is the perpendicular bisector of segment YS. Line O is the reflection of line W across point R. Point Q is the reflection of point R across line O. Find the distance between points Q and Y. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FK = 3. Line J passes through points K and F. Let Y be the rotation of point F by angle 3pi/4 around point K. Let S be the line through point K perpendicular to line J. Line A is the reflection of line S across point F. Point V is the reflection of point Y across line A. Point M is the reflection of point V across line J. Let Q be the midpoint of segment YM. What is the distance between points M and Q? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ET = 11. Let F be the rotation of point T by angle 2pi/3 around point E. Point O is the midpoint of segment FT. Line B is the perpendicular bisector of segment FT. Point A is the midpoint of segment EO. Point U is obtained by rotating point E by angle pi/4 about point A. Point G is obtained by rotating point T by angle pi/6 about point E. Line Y passes through point U parallel to line B. Point I is the reflection of point G across line Y. Let N be the midpoint of segment OI. Let H be the rotation of point E by angle pi around point I. Point Z is the midpoint of segment AT. Point K is the midpoint of segment HA. Let C be the rotation of point T by angle pi/2 around point A. Line Q is the perpendicular bisector of segment ZN. Let J be the reflection of line Q through point C. Point X is the reflection of point K across segment UI. Point D is the intersection of lines Y and J. What is the length of segment XD? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
MP = 3. Let D be the vector from point M to point P. Let G be the rotation of point P by angle pi/3 around point M. Point C is the reflection of point G across segment MP. Point W is the translation of point C by vector D. Let E be the rotation of point P by angle 5pi/6 around point M. Let A be the line through points W and G. Point O is the reflection of point M across point E. Line I is the reflection of line A across point O. Point L is the reflection of point G across line I. Find the distance between points L 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 DP = 3. Point U is the reflection of point D across point P. Let L be the line through points D and U. Let S be the line through point D perpendicular to line L. Point H is the reflection of point P across line S. What is 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]
UR = 3. Point E is obtained by rotating point U by angle 5pi/6 about point R. Point Z is the midpoint of segment UE. Let A be the perpendicular bisector of segment UR. Point N is the midpoint of segment UR. Let X be the line through point Z parallel to segment UR. Let P be the reflection of point N across point U. Point T is the reflection of point P across line X. Point M is obtained by rotating point T by angle pi/2 about point R. Let D be the line through point R parallel to line A. Point W is the intersection of line D and segment EP. Point V is obtained by rotating point M by angle pi about point R. Point H is the reflection of point R across segment EP. Let J be the reflection of point V across point E. Point G is the reflection of point T across segment WJ. What is the length of segment HG? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XM = 6. Let L be the rotation of point M by angle pi/4 around point X. Let Q be the midpoint of segment LX. Let N be the line through point Q perpendicular to segment MX. Point P is the reflection of point L across line N. Let T be the midpoint of segment MX. Find the distance between points T and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
UK = 9. Let V be the reflection of point U across point K. Let Q be the reflection of point V across point U. Point E is the reflection of point K across point V. Point N is obtained by rotating point V by angle 7pi/12 about point E. Let R be the midpoint of segment QK. Let J be the perpendicular bisector of segment KU. Let W be the perpendicular bisector of segment QK. Let S be the rotation of point K by angle 11pi/12 around point R. Let Z be the midpoint of segment RN. Line T passes through point N parallel to line W. Let H be the reflection of point Z across point R. Line Y passes through point S parallel to line T. Line A is the reflection of line Y across line J. Line O is the reflection of line A across line W. Point B is the reflection of point H across line O. What is 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 KP = 7. Let B be the perpendicular bisector of segment PK. Let U be the reflection of line B through point P. Point Q is the reflection of point K across point P. Line E is the reflection of line B across line U. Point C lies on segment PK. Line I passes through point K perpendicular to segment PK. Point H is the midpoint of segment PK. Point N is the reflection of point H across line E. Point T is the reflection of point N across line U. Line J is the reflection of line E across line I. Line M passes through point Q parallel to line U. Line F is the reflection of line J across point C. Point V lies on line M. Let O be the reflection of point T across point V. Let G be the line through point O parallel to line F. Point X is the reflection of point T across line G. Find the distance between points K and X. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let KL = 6. Let P be the midpoint of segment KL. Let U be the perpendicular bisector of segment KL. Let N be the reflection of point L across point K. Point R is on line U with |RP| = 6. Let O be the reflection of point R across point N. Let I be the midpoint of segment PL. Let G be the midpoint of segment LI. Line C passes through point O perpendicular to segment KL. Point F is the reflection of point G across line C. Find the distance between points P and F. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EN = 12. Let M be the reflection of point E across point N. Let J be the rotation of point M by angle 3pi/4 around point N. Let W be the midpoint of segment JN. Point Y is the midpoint of segment WM. Point B is the reflection of point E across point Y. Point O is the midpoint of segment NB. Point G is the midpoint of segment BM. What is the distance between points G and O? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VP = 4. Let H be the rotation of point V by angle pi/3 around point P. Let Q be the midpoint of segment HV. Let L be the midpoint of segment PQ. Let J be the perpendicular bisector of segment HP. Let R be the reflection of line J through point L. Point A is the midpoint of segment HP. Let E be the rotation of point Q by angle 7pi/12 around point P. Segment VP and line R intersect at point X. Point Z is the reflection of point X across point E. Let T be the reflection of point V across point Z. Let O be the angle bisector of angle TPQ. Point D is the reflection of point A across line O. What is the distance between points D and T? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ES = 3. Line U is the perpendicular bisector of segment ES. Let K be the line through point S parallel to line U. Let B be the reflection of line K across line U. Line Z is the reflection of line B across point S. Point C lies on line B at distance 8 from point E. Let L be the midpoint of segment SE. Line F passes through points C and L. Point A is the reflection of point S across line B. Lines F and Z intersect at point D. Let O be the midpoint of segment DA. Find the length of segment EO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GA = 9. Let D be the midpoint of segment GA. Let S be the rotation of point D by angle 3pi/4 around point A. Let J be the reflection of point S across point D. Let V be the perpendicular bisector of segment AS. Line N is the perpendicular bisector of segment DS. Let W be the line through point J parallel to line N. Line E is the reflection of line N across line V. Lines W and E intersect at point L. What is the length of segment LG? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WO = 5. Let D be the reflection of point O across point W. Point R is obtained by rotating point W by angle 5pi/6 about point D. Point Y is the reflection of point R across point D. Point H is the reflection of point D across point Y. Point T is the reflection of point W across point H. Point Q is the reflection of point D across segment WT. Find 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]
Let DY = 4. Let B be the midpoint of segment YD. Let T be the rotation of point Y by angle 5pi/6 around point B. Let N be the perpendicular bisector of segment DY. Let S be the rotation of point B by angle pi/3 around point Y. Point K is the reflection of point S across line N. Let Q be the reflection of line N through point S. Point M is the reflection of point K across line Q. Point A is the reflection of point T across line N. 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]
LI = 11. Let D be the reflection of point L across point I. Point X is the midpoint of segment LI. Let A be the rotation of point D by angle 2pi/3 around point L. Point O is obtained by rotating point D by angle pi about point A. Let T be the line through point O perpendicular to segment LI. Point K is the midpoint of segment XO. Point F is the reflection of point D across line T. Point G is the reflection of point K across segment LI. Point H is the reflection of point A across point G. Find the distance between points F 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 LP = 12. Point N is obtained by rotating point P by angle pi/12 about point L. Let I be the rotation of point P by angle pi/2 around point L. Let T be the midpoint of segment LP. Point U is the reflection of point T across segment PI. Point R is the reflection of point U across segment LP. Find the distance between points N 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 PI = 10. Line W passes through point P perpendicular to segment IP. Point H is the reflection of point I across line W. Let Q be the reflection of line W through point I. Point R is the reflection of point H across line Q. Find the distance between points H 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 GO = 2. Let S be the reflection of point G across point O. Let C be the line through points S and G. Point Y is obtained by rotating point S by angle pi/12 about point G. Let B be the angle bisector of angle SOY. Let Z be the line through point O perpendicular to line C. Let V be the reflection of line Z across line B. Point K is the reflection of point Y across line V. Line E passes through point G parallel to line B. Point J is the midpoint of segment GK. Segment YJ and line E intersect at point X. Find the distance between points Y and X. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JR = 6. Z is placed 11 units away from R. Line D passes through points R and Z. Let W be the midpoint of segment RZ. Let N be the vector from point Z to point W. Point X is obtained by rotating point Z by angle 11pi/12 about point W. Let F be point R translated by vector N. Point S is the reflection of point F across segment ZX. Point T is the midpoint of segment RW. Point E is the reflection of point T across segment ZX. Point K is the reflection of point E across line D. Let V be the reflection of point W across point S. Find the length of segment VK. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let HN = 5. Point I is the midpoint of segment NH. Line G is the angle bisector of angle HIN. Line S is the perpendicular bisector of segment IN. Point V is the reflection of point I across point N. Point R is on line S with |RN| = 5. Let Q be the reflection of point H across point R. Point U is the reflection of point Q across line G. Point T is the reflection of point U across line S. Point Z is the reflection of point T across line G. Find the distance between points V and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let HF = 1. Let X be the midpoint of segment FH. Let M be the midpoint of segment XH. Let L be the perpendicular bisector of segment MX. Point K is on line L with |KM| = 2. Point B is the midpoint of segment KH. Let N be the line through point B parallel to line L. Point Y is the reflection of point M across line N. Point I is the reflection of point F across segment BY. What is the length of segment YI? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LX = 11. Point T is obtained by rotating point X by angle 7pi/12 about point L. Let W be the vector from point T to point L. Let K be point X translated by vector W. Point O is obtained by rotating point L by angle 3pi/4 about point T. Let Z be the angle bisector of angle TXL. Let G be the rotation of point O by angle 5pi/12 around point K. Point A is the midpoint of segment TO. Let C be the midpoint of segment GK. Let H be point A translated by vector W. Point F is the midpoint of segment CT. Line Y passes through points X and A. Point M is the reflection of point F across point H. Let V be the reflection of line Z through point F. Point E is the reflection of point M across point G. Lines Y and V intersect at point N. Find the length of segment NE. Express your answer as a decimal number rounded to 2 decimal places.

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


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HM = 8. Point D is obtained by rotating point M by angle pi/4 about point H. Let V be the vector from point M to point H. Let U be point D translated by vector V. Let I be the midpoint of segment HU. Point O is the reflection of point U across point D. Let Z be the reflection of point O across point I. Point G is the midpoint of segment MH. What is the distance between points Z and G? Express your answer as a decimal number rounded to 2 decimal places.

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