, 2021

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.