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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 $PQ = 36,$ $PR = 22,$ and $MY = 8,$ then find the area of triangle $PYR.$

 Apr 11, 2023
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The perpendicular bisector of PQ divides PQ into two segments of equal length, so QM = QY = 18. Since PR bisects ∠QPR, we have ∠QPR=∠RPQ=45∘. Therefore, △PQR is a 45-45-90 right triangle, so PY=QYsqrt(2)​=18sqrt(2)​.

The area of a triangle is given by the formula 1/2*​bh, where b is the base and h is the height. In this case, the base of △PYR is PQ = 36, and the height is PY = 18\sqrt{2}, so the area is

1/2​(36)(18*sqrt(2)​)=288*sqrt(2).

 Apr 11, 2023

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