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.

kittykat Dec 17, 2023

#1**0 **

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=2DE=52.

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

BuiIderBoi Dec 17, 2023