Let's call the perpendicular bisector of PQ as line segment ZY. Since M is the midpoint of PQ, it follows that PZ = ZQ = 18.
We can now use the Pythagorean theorem to find the length of PX and QX:
PX^2 + MY^2 = PQ^2 (since PX is the angle bisector)
PX^2 + 8^2 = 36^2
PX^2 = 36^2 - 8^2
PX^2 = 1296
PX = 36
Similarly,
QX^2 + MY^2 = QR^2
QX^2 + 8^2 = 22^2
QX^2 = 22^2 - 8^2
QX^2 = 484
QX = 22
Since PX is an angle bisector, it follows that PR/PX = QR/QX
PR/PX = 22/36
QR/QX = 36/22
QR = 22 * (36/22) = 36
Since QX + PX = QR = 36, it follows that PX = 36 - QX = 36 - 22 = 14
Finally, we can use the formula for the area of a triangle:
Area = (1/2) * PY * RX
Area = (1/2) * 8 * 14
Area = 56
Therefore, the area of triangle PYR is 56.
