2006 AMC 12A Problems/Problem 16
Problem
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Circles with centers 
and 
have radii 
and 
, respectively. A common internal tangent intersects the circles at 
and 
, respectively. Lines 
and 
intersect at 
, and 
. What is ?
Solution
and 
(vertical angles) are congruent, as are right angles 
and 
(since radii intersect tangents at right angles). Thus, 
.
By the Pythagorean Theorem, line segment 
. The sides are proportional, so 
. This makes 
and 
.
See also
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| Preceded by Problem 15 |
AMC 12A 2006 |
Followed by Problem 17 |
