Math Prize 2012 Problem 5
Source:
September 24, 2012
geometrysimilar triangles
Problem Statement
The figure below shows a semicircle inscribed in a right triangle.
[asy]
draw((0, 0) -- (15, 0) -- (0, 8) -- cycle);
real r = 120 / 23;
real theta = -aTan(8/15);
draw(arc((r, r), r, theta + 180, theta + 360));
[/asy]
The triangle has legs of length 8 and 15. The semicircle is tangent to the two legs, and its diameter is on the hypotenuse. What is the radius of the semicircle?