1959 AMC 12 Problem 41
Attempt Problem 41 of the 1959 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1959 AMC 12 solutions, or check the answer key.
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41.
On the same side of a straight line three circles are drawn as follows: a circle with a radius of inches is tangent to the line, the other two circles are equal, and each is tangent to the line and to the other two circles. The radius of the equal circles is:
Answer: D
Small Hint:
The small circle lies symmetrically between the two equal circles
Big Hint:
If an equal circle has radius compare the center distance with
Solution:
Let the equal circles have radius Their centers are apart, so the center of the radius- circle lies midway between them. The horizontal and vertical separations between its center and either large center are and Tangency gives Simplifying yields and positivity gives
Thus, the correct answer is D.