2018 AMC 10A Problem 15

Question

Two circles of radius

5

are externally tangent to each other and are internally tangent to a circle of radius

13

at points

A

and

B,

as shown in the diagram. The distance

AB

can be written in the form

\frac{m}{n},

where

m

and

n

are relatively prime positive integers. What is

m+n?

21

29

58

69

93

Join the three circle centers

The triangle through tangent points is similar to the triangle through centers

Let

X

be the center of the large circle and let

Y,Z

be the centers of the two smaller circles. Then

XY=XZ=13-5=8

and

YZ=10

.

The radii to tangent points make

XA

collinear with

XY

and

XB

collinear with

XZ

, so

\triangle XAB\sim \triangle XYZ

. Thus

\dfrac{AB}{YZ}=\dfrac{XA}{XY}=\dfrac{13}{8}

.

Hence

AB=10\cdot\dfrac{13}{8}=\dfrac{65}{4}

, so

m+n=65+4=69

. Thus, D is the correct answer.

Problem 15 in other years