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 $PYR.$
Let's first find the length of PX. Since PX bisects angle QPR, we have:
PQ / PX = PR / QR
We can simplify this using the lengths of PQ and PR that we know:
36 / PX = 22 / QR
Solving for PX:
PX = (36 * QR) / 22
Since M is the midpoint of PQ, we have:
PQ = 2 * MY = 2 * 8 = 16
So QR can be found as:
QR = PR + PQ = 22 + 16 = 38
Substituting into our expression for PX:
PX = (36 * 38) / 22 = 54
Next, let's find the length of PY. Since MY = 8, we can use the Pythagorean Theorem to find PY:
PY^2 + MY^2 = PX^2 PY^2 + 8^2 = 54^2 PY^2 = 54^2 - 8^2 = 2916 PY = sqrt(2916) = 54
Finally, we can use the formula for the area of a triangle:
Area = (1/2) * PY * PX Area = (1/2) * 54 * 54 Area = 1458
Therefore, the area of triangle PYR is 1458.
