1976 AHSME Problems/Problem 26
Problem
In the adjoining figure, every point of circle is exterior to circle . Let and be the points of intersection of an internal common tangent with the two external common tangents. Then the length of is
Solution
By the Two Tangent Theorem, the segments drawn from point tangent to circle are congruent. Let them have length . Similarly, the two segments drawn from point tangent to circle are congruent. Let them have length .
Let the length of the two external common tangents be . We know that the two segments drawn from point tangent to circle are congruent. One of these tangents (the one on the external common tangent) has length , so the other (the one on the internal common tangent) also has length Doing the same thing for point and circle shows that the segments drawn from tangent to circle has length . Thus, , because the numerator of this fraction counts the length of twice. Because is the length of the external common tangent, our answer is .
See Also
1976 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 25 |
Followed by Problem 27 | |
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