Question

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

$13$

$\dfrac{44}{3}$

$\sqrt{221}$

$\sqrt{255}$

$\dfrac{55}{3}$

Answer: B

Concepts:tangent line similarity Pythagorean Theorem

Difficulty rating: 1720

Solution:

Since $AC \perp CD,$ we have $CE = \sqrt{AE^2 - AC^2}$ $= \sqrt{25 - 9}$ $= 4.$

Because $\triangle ACE \sim \triangle BDE,$ $\frac{DE}{CE} = \frac{BD}{AC},$ so $DE = 4 \cdot \frac{8}{3} = \frac{32}{3}.$

Then $CD = CE + DE$ $= 4 + \frac{32}{3}$ $= \frac{44}{3}.$

Thus, the correct answer is B.

2006 AMC 10A Problem 23

Answer: B

Difficulty rating: 1720

Solution:

Problem 23 in Other Years