2006 AMC 12A Problems/Problem 16

Problem

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Circles with centers $A$ and $B$ have radii $3$ and $8$ , respectively. A common internal tangent intersects the circles at $C$ and $D$ , respectively. Lines $AB$ and $CD$ intersect at $E$ , and $AE=5$ . What is $CD$ ?

$\mathrm{(A) \ } 13\qquad \mathrm{(B) \ } \frac{44}{3}\qquad \mathrm{(C) \ } \sqrt{221}\qquad \mathrm{(D) \ } \sqrt{255}$ $\mathrm{(E) \ } \frac{55}{3}$

Solution

$\angle AEC$ and $\angle BED$ (vertical angles) are congruent, as are right angles $\angle ACE$ and $\angle BDE$ (since radii intersect tangents at right angles). Thus, $\triangle ACE \sim \triangle BDE$ .

By the Pythagorean Theorem, line segment $CE = 4$ . The sides are proportional, so $\frac{CE}{AC} = \frac{DE}{BD} \Rightarrow \frac{4}{3} = \frac{DE}{8}$ . This makes $DE = \frac{32}{3}$ and $CD = CE + DE = 4 + \frac{32}{3} = \frac{44}{3} \Rightarrow \mathrm{B}$ .

2006 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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2006 AMC 12A Problems/Problem 16

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