Points X, Y, and Z are on the sides QR , PR, and PQ respectively, of right triangle PQR such that PZXY is a square. If PQ=10 and PR=10, then what is the side length of the square?
Let's denote the side length of the square as s. Since PZXY is a square, we know that PZ = ZX = s, and QX = RY = 10 - s (since QX and RY are the remaining segments of length 10 after subtracting the side length of the square).
Using the Pythagorean theorem, we can set up two equations based on the right triangles PZQ and PXR:
s^2 + QX^2 = QP^2 (1) s^2 + RY^2 = RP^2 (2)
Substituting in the values we know:
s^2 + (10 - s)^2 = 100 (1) s^2 + (10 - s)^2 = 100 (2)
Expanding and simplifying:
2s^2 - 20s + 100 = 100 2s^2 - 20s = 0 2s(s - 10) = 0
This gives us two solutions: s = 0 (which doesn't make sense since the square has positive length) and s = 10.
Therefore, the side length of the square is 10.