Problem:
A square with side length is inscribed in a right triangle with sides of length 3,4 , and 5 so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length is inscribed in another right triangle with sides of length 3,4 , and 5 so that one side of the square lies on the hypotenuse of the triangle. What is
Answer Choices:
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Solution:
In the first figure , so and . In the second figure, the small triangles are similar to the large one, so the lengths of the portions of the side of length 3 are as shown. Solving yields . Thus .
The problems on this page are the property of the MAA's American Mathematics Competitions