$\triangle WYZ$ is equilateral and $\triangle WXY$ is isosceles with a vertex angle of $\angle XWY.$ If the perimeter of the entire figure is $22$ units and the area of the entire figure is $9 \sqrt{3} + 4 \sqrt{5},$ what are the side lengths of each side of each triangle?

Pangolin14 Nov 3, 2020