Difference between revisions of "1959 AHSME Problems/Problem 41"
Wintermath (talk | contribs) (Created page with "On the same side of a straight line three circles are drawn as follows: a circle with a radius of <math>4</math> inches is tangent to the line, the other two circles are equal...") |
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| + | == Problem == | ||
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On the same side of a straight line three circles are drawn as follows: a circle with a radius of <math>4</math> 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: <math>\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 12</math> | On the same side of a straight line three circles are drawn as follows: a circle with a radius of <math>4</math> 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: <math>\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 12</math> | ||
| − | Solution | + | == Solution == |
Make a line DG which is parallel to FC. We know that GF = BF = 5, since BFE is similar to BGD. | Make a line DG which is parallel to FC. We know that GF = BF = 5, since BFE is similar to BGD. | ||
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Since ADG is similar to ACB, we know that AG is 10. | Since ADG is similar to ACB, we know that AG is 10. | ||
| − | Thus, AF = 10 + 5 = 15, which is answer C. | + | Thus, AF = 10 + 5 = 15, which is answer <math>\boxed{C}</math>. |
Revision as of 14:05, 16 July 2024
Problem
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: 
Solution
Make a line DG which is parallel to FC. We know that GF = BF = 5, since BFE is similar to BGD.
Since ADG is similar to ACB, we know that AG is 10.
Thus, AF = 10 + 5 = 15, which is answer 
.
