In the diagram, each of $\triangle W X Z$ and $\triangle X Y Z$ is an isosceles right-angled triangle. The length of $W X$ is $6 \sqrt{2}$. The perimeter of quadrilateral $W X Y Z$ is closest to
(A) 18
(B) 20
(C) 23
(D) 25
(E) 29

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-015.jpg?height=396&width=244&top_left_y=1168&top_left_x=1298)
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 $\triangle W Z X$ which is isosceles and right-angled at $Z$.

Here, $W X: W Z=\sqrt{2}: 1$. Since $W X=6 \sqrt{2}$, then $W Z=\frac{6 \sqrt{2}}{\sqrt{2}}=6$.

Since $\triangle W Z X$ is isosceles, then $X Z=W Z=6$.

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

Here, $Y Z=\frac{X Z}{\sqrt{2}}=\frac{6}{\sqrt{2}}=\frac{6}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{6 \sqrt{2}}{2}=3 \sqrt{2}$.

Since $\triangle X Y Z$ is isosceles, then $X Y=Y Z=3 \sqrt{2}$.

Therefore, the perimeter of $W X Y Z$ is

$$
W X+X Y+Y Z+W Z=6 \sqrt{2}+3 \sqrt{2}+3 \sqrt{2}+6=12 \sqrt{2}+6 \approx 22.97
$$

Of the given choices, this is closest to 23 .

ANswer: (C)