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 $m+n?$

$21$

$29$

$58$

$69$

$93$

Solution:

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.

2018 AMC 10A Problem 15

Solution:

Problem 15 in Other Years