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Geometry - pls help i don't understand how to do it

+2
8
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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
+9665
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Note that PX bisects $$\angle QPR$$. Then, we can suppose $$\angle XPQ = \angle XPR = \theta$$.

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 $$\phi$$, then the area of the triangle is $$\dfrac12 ab \sin \phi$$.

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 $$\dfrac12 \cdot 8 \cdot 18 = 72$$. But on the other hand, using the formula stated above, the same area is also equal to $$\dfrac12 (PM)(PY) \sin \theta = 9\cdot PY \sin \theta$$.

Therefore, we know that $$PY \sin \theta = 8$$ by comparing.

Now, we look at triangle PYR. By the formula, the area is $$\dfrac12 (PY)(PR) \sin \theta$$, but we know PR = 22. Using $$PY \sin \theta = 8$$, the required area is $$\dfrac12 (22) (PY \sin \theta) = 11(8) = \boxed{88}$$.

Apr 30, 2024
#2
+46
+1

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

Apr 30, 2024