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?

Answer 1

Since PZXY is a square, we know that QZ = QX and RY = RX. Let's call the side length of the square s. Then, we have:

QZ + RY = PQ - s (since QZ and RY are on the sides PQ and PR, respectively)
QX + RX = QR - s (since QX and RX are on the side QR)

But we know that PQ = 10 and PR = 10, so PQ - s = PR - s = 10 - s. Therefore:

QZ + RY = 10 - s
QX + RX = QR - s

We also know that QZ + QX = QP and RY + RX = RP, so:

QP + RP = QZ + QX + RY + RX = (QZ + RY) + (QX + RX) = (10 - s) + (QR - s)

But we also know that PQ^2 + PR^2 = QR^2 (since PQR is a right triangle), so:

10^2 + 10^2 = QR^2
QR = sqrt(200) = 10sqrt(2)

Substituting this into the equation above, we have:

QP + RP = (10 - s) + (10sqrt(2) - s)

Since QP + RP = PQ + PR = 20, we can solve for s:

20 = (10 - s) + (10sqrt(2) - s)
20 = 20sqrt(2) - 2s
2s = 20sqrt(2) - 20
s = 10sqrt(2) - 10

Therefore, the side length of the square is s = 10sqrt(2) - 10.