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Problem 557

AMC 10/12, early questions

Geometry Difficulty 3.5 Find the answer CEMC Fermat · Canada · 2021

In the diagram, each of $\triangle WXZ$ and $\triangle XYZ$ is an isosceles right-angled triangle.

The length of $WX$ is $6\sqrt 2$ . The perimeter of quadrilateral $WXYZ$ is closest to

$18$
$20$
$23$
$25$
$29$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

In an isosceles right-angled triangle, the ratio of the length of the hypotenuse to the length of each of the shorter sides is $\sqrt{2}:1$ .

Consider $\triangle WZX$ which is isosceles and right-angled at $Z$ .

Here, $WX:WZ = \sqrt{2}:1$ . Since $WX=6\sqrt{2}$ , then $WZ = \dfrac{6\sqrt{2}}{\sqrt{2}}=6$ .

Since $\triangle WZX$ is isosceles, then $XZ = WZ = 6$ .

Consider $\triangle XYZ$ which is isosceles and right-angled at $Y$ .

Here, $YZ = \dfrac{XZ}{\sqrt{2}} = \dfrac{6}{\sqrt{2}} = \dfrac{6}{\sqrt{2}} \cdot \dfrac{\sqrt{2}}{\sqrt{2}} = \dfrac{6\sqrt{2}}{2} = 3\sqrt{2}$ .

Since $\triangle XYZ$ is isosceles, then $XY = YZ = 3\sqrt{2}$ .

Therefore, the perimeter of $WXYZ$ is $WX + XY + YZ + WZ = 6\sqrt{2}+3\sqrt{2}+3\sqrt{2}+6 = 12\sqrt{2}+6 \approx 22.97$ Of the given choices, this is closest to 23.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.