smc

Question

A square with side length x 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 y 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 xy?

Accepted Answer

Analyze the first right triangle. FIGURE Note that ABC and FBE are similar, so BFFE = ABAC. This can be written as 4-xx=43. Solving, x = 127. Now we analyze the second triangle. FIGURE Similarly, A'B'C' and RB'Q are similar, so RB' = 43y, and C'S = 34y. Thus, C'B' = C'S + SR + RB' = 43y + y + 34y = 5. Solving for y, we get y = 6037. Thus, xy = 3735 3735.