Problem:
A circle of radius 1 is tangent to a circle of radius 2. The sides of â–³ABC are tangent to the circles as shown, and the sides AB and AC are congruent. What is the area of â–³ABC?
Answer Choices:
A. 235​
B. 152​
C. 364​
D. 162​
E. 24
Solution:
Let O and O′ denote the centers of the smaller and larger circles, respectively. Let D and D′ be the points on AC that are also on the smaller and larger circles, respectively. Since △ADO and △AD′O′ are similar right triangles, we have
1AO​=2AO′​=2AO+3​, so AO=3
As a consequence,
AD=AO2−OD2​=9−1​=22​
Let F be the midpoint of BC. Since â–³ADO and â–³AFC are similar right triangles, we have
.jpg)
