Question

Isosceles right triangle $ABC$ encloses a semicircle of area $2\pi.$ The circle has its center $O$ on hypotenuse $\overline{AB}$ and is tangent to sides $\overline{AC}$ and $\overline{BC}.$ What is the area of triangle $ABC?$

$6$

$8$

$3 \pi$

$10$

$4 \pi$

Solution:

Reflect the triangle and semicircle across hypotenuse $\overline{AB}$ . This forms a full circle inscribed in a square.

The semicircle has area $2\pi$ , so the full circle has area $4\pi$ and radius $2$ . The square side length is therefore $4$ .

The square area is $16$ , so the original triangle has half that area, $8$ .

Thus, B is the correct answer.

2005 AMC 8 Problem 23

Solution:

Problem 23 in Other Years