2006 AMC 10A Problems/Problem 23
Problem
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Circles with centers and
have radii 3 and 8, respectively. A common internal tangent intersects the circles at
and
, respectively. Lines
and
intersect at
, and
. What is
?
Solution
and
are vertical angles so they are congruent, as are angles
and
(both are right angles because the radius and tangent line at a point on a circle are always perpendicular). Thus,
.
By the Pythagorean Theorem, line segment . The sides are proportional, so
. This makes
and
.
See also
2006 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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All AMC 10 Problems and Solutions |