I woule like a proper explaination please! Not just the answer!

In triangle \(PQR, M\) is the midpoint of \( \overline{PQ}\) Let \(X\) be the point on \(\overline{QR}\) such that \(\overline{PX}\) bisects \(\angle{QPR}\) and let the perpendicular bisector of \(\overline{PQ}\) intersect \(\overline{PX}\) at \(Y\) If \(PQ = 36\) and \(MY = 8\) then find the area of triangle \(PYR.\)

icecreamlover Jul 5, 2023

#1**-1 **

Here are the steps to solve the problem:

Since M is the midpoint of PQ, then QM = 18.

Since PX bisects angle QPR, then PX = QR/2 = 19.

Since Y is on the perpendicular bisector of PQ, then PY = QY = 18.

Since MY = 8, then PY = 18 - 8 = 10.

Since PX = 19 and PY = 10, then triangle PXY is a 9-10-11 right triangle.

Since the perpendicular bisector of PQ intersects PX at Y, then PY is an altitude of triangle PQR.

Therefore, the area of triangle PYR is (1/2)(PQ)(PY) = (1/2)(36)(10) = 180.

Therefore, the area of triangle PYR is 180.

Guest Jul 5, 2023