2012 AMC 10A Problem 11

Problem 11 of 25IntermediateGeometry

Externally tangent circles with centers at points

A

and

B

have radii of lengths

5

and

3,

respectively. A line externally tangent to both circles intersects ray

AB

at point

C.

What is

BC?

Show solution

Solution

Let

x

BC.

Note that

\triangle CEB

and

\triangle CDA

are similar due to angle-angle (tangent lines are perpendicular to radii). Then

\dfrac{x}{3} = \dfrac{8 + x}{5}.

Cross-multiplying gives us

5x = 24 + 3x

x = 12.

Thus, D is the correct answer.

Tagged: tangent circles · tangent line · similarity