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A semicircle is inscribed in triangle $XYZ$ so that its diameter lies on $\overline{YZ}$, and is tangent to the other two sides.   If $XY = 10,$ $XZ = 10,$ and $YZ = 10,$ then find the area of the semicircle.

 Dec 17, 2023
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Let O be the center of the semicircle, and let D and E be the points of tangency with XY and XZ, respectively. Since the semicircle's diameter lies on YZ, we have D=X and E=Z.

 

Therefore, triangle DEO is a right triangle with hypotenuse DE=YZ=10.

 

 

 

Since XOE is a diameter, ∠XOE=90∘. Because of the tangency conditions, ∠DOE=∠EOX=45∘, so triangle DEO is a 45-45-90 triangle.

 

Therefore, DO=EO=2​DE​=52​​.

 

The area of the semicircle is then 21​⋅π⋅(DO)2=21​⋅π⋅(52​)2=50π​.

 Dec 17, 2023

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