Chapter 5.1, Problem 46MS

Here we celebrate the power of algebra as a powerful way of finding unknown quantities by naming them, of expressing infinitely many relationships and connecl ions clearly and succinctly, and of uncovering pattern and structure.

Ratio-rama. These two shapes are equivalent by distortion. (Do you see why?) The left shape has squares for its boundaries and the right shape has isosceles right triangles for its boundaries. Let x denote the side length of the inner square and y denote the base (and height) of the inner triangle. If the side length of the outer square is $40x$ and the base (and height) of the outer triangle is $3y$, find the ratio of the area of the outer square to the area of the inner square and the ratio of the area of the outer triangle to the area of the inner triangle.