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Question

There are two natural ways to inscribe a square in a given isosceles right triangle. If it is done as in Figure 1 below, then one finds that the area of the square is 441 cm^2. What is the area (in cm^2) of the square inscribed in the same ABC as shown in Figure 2 below? FIGURE

Accepted Answer

We are given that the area of the inscribed square is 441, so the side length of that square is 21. Since the square divides the 45-45-90 larger triangle into 2 smaller congruent 45-45-90, then the legs of the larger isosceles right triangle (BC and AB) are equal to 42. FIGURE We now have that 3S=422, so S=142. But we want the area of the square which is S^2=(142)^2= 392