Question

Three circular arcs of radius $5$ units bound the region shown. Arcs $AB$ and $AD$ are quarter-circles, and arc $BCD$ is a semicircle. What is the area, in square units, of the region?

$25$

$10+5\pi$

$50$

$50+5\pi$

$25\pi$

Solution:

Create a rectangle that covers the bottom half of the figure as shown below.

Then, we get that $[ABCD] = [ABD] + [BCD].$

We also know that $[ABD] = [BDEF] - [ABE]$ $- [ADF].$

$ABE$ and $ADF$ are both quartercircles that form a semicircle with the same area as $CBD.$

This means that $[ABD] = [BDEF] - [BCD]$ and $[ABCD] = [BCD] + [BDEF]$ $- [BCD]$ $= [BDEF] = 10 \cdot 5 = 50.$

Thus, C is the correct answer.

2000 AMC 8 Problem 19

Solution:

Problem 19 in Other Years