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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+9676 
+4

Note that PX bisects QPR. Then, we can suppose .

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 , then the area of the triangle is .

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 . But on the other hand, using the formula stated above, the same area is also equal to .

Therefore, we know that  by comparing.

 

Now, we look at triangle PYR. By the formula, the area is , but we know PR = 22. Using , the required area is .

 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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