1961 AHSME Problems/Problem 11
Problem
Two tangents are drawn to a circle from an exterior point ; they touch the circle at points and respectively. A third tangent intersects segment in and in , and touches the circle at . If , then the perimeter of is
Solution
Draw the diagram as shown. Note that the two tangent lines from a single outside point of a circle have the exact same length, so , , and .
The perimeter of the triangle is . Note that , so from substitution, the perimeter is Thus, the perimeter of the triangle is , so the answer is .
Solution 2 (Non-rigorous)
Since can be anywhere on the circle between and , it can basically be "on top" of . Then will be at the same point as , so form a degenerate triable with side length . So its perimeter will be . Since and by power of a point, as and decrease in length, will "grow" to compensate, so the perimeter will stay constant with a value of .
We can also skip verifying that the perimeter will stay constant, since it seems unlikely that MAA would create a question with as the answer.
~jd9
See Also
1961 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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