+1725 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 $YZ = 10 \sqrt{2},$ then find the area of the semicircle.
+129771 Since YZ =10 sqrt 2 and XY = XZ = 10....then XYZ is a right triangle with the right angle at X
Call the center of the semi-circle O
Let the tangent XZ intersect the circle at P
Draw OX.....this bisects angle YXZ
So we have right triangle POX
Angle OXP = 45
Angle OPX = 90
So angle XOP = 45
So triangle OPX is a 45 -45- 90 right triangle with OX = (1/2)YZ = (1/2) 10sqrt (2) = 5sqrt2
And OP = XP = 5sqrt 2 / sqrt 2 = 5
So OP is the radius of the semi-circle
Area of semi-circle = (1/2) pi * (5)^2 = 12.5 pi ≈ 39.27
