A house? or not

Geometry Level 2

Triangle $ABC$ is a right triangle with $\angle A = 90^\circ$ . Point $W$ on segment $AB$ , $X$ on $AC$ , and points $Y$ and $Z$ on $BC$ are such that $WXYZ$ is a square.

If $BZ = 6$ and $YC = 8$ (the remaining $YZ$ is a side of the square), find the area of the square $WXYZ$ .

The answer is 48.

2 solutions

Chew-Seong Cheong
May 28, 2021

Let the side length of square $WXYZ$ be $a$ . Then the area of square $WXYZ$ is $\blue{a^2}$ . We note that $\triangle CXY$ and $\triangle BWZ$ are similar and $\angle XCY = \angle BWZ = \theta$ . Then

$\tan \theta = \frac a8 = \frac 6a \implies \blue{a^2} = \boxed{48}$

Saya Suka
May 27, 2021

Should be "If BZ = 6 and YC = 8" in the problem statement. Anyway, we have the ratio
BZ / ZW = XY / YC
6 / s = s / 8
s² = 6 × 8 = 48

A house? or not

The answer is 48.

2 solutions

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