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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 30, 2024
 #1
avatar+9675 
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Note that PX bisects QPR. Then, we can suppose XPQ=XPR=θ.

Recall the formula:

If two sides of a triangle are of length a and b respectively, and the angle included by these two sides is ϕ, then the area of the triangle is 12absinϕ.

Now, draw the diagram and look at triangle PMY. Its height is MY (which is 8) and its base is PM (which is 18). So the area of triangle PMY is 12818=72. But on the other hand, using the formula stated above, the same area is also equal to 12(PM)(PY)sinθ=9PYsinθ.

Therefore, we know that PYsinθ=8 by comparing.

 

Now, we look at triangle PYR. By the formula, the area is 12(PY)(PR)sinθ, but we know PR = 22. Using PYsinθ=8, the required area is 12(22)(PYsinθ)=11(8)=88.

 Apr 30, 2024
 #2
avatar+71 
+4

I managed to solve it myself and got the same thing!!

 Apr 30, 2024

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