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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 $PYR.$ Include explanation pls
 Feb 5, 2023
 #1
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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.

 Feb 5, 2023
 #2
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Thank you, that's correct!

Guest Feb 12, 2023

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