2012 AMC 10A Problem 11

Question

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?

4

4.8

10.2

12

14.4

Draw radii to the two tangent points

The two right triangles with vertex

C

are similar

Let

x

be

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.

Problem 11 in other years