Inside a square with side length , two congruent equilateral triangles are drawn such that they share one side and each has one vertex on a vertex of the square. What is the side length of the largest square that can be inscribed in the space inside the square and outside of the triangles?

gb1falcon May 2, 2023

#1**-2 **

Let O be the center of the square, and let E, F, G, and H be the points where the sides of the square intersect the sides of the triangles. Since the triangles are equilateral, OE=OF=OG=OH=5. Let s be the side length of the largest square that can be inscribed in the space inside the square and outside of the triangles. Since O is the center of the square, s must be less than 5.

Let x=OE−s. Since s is the side length of the inscribed square, OE+s=OE−x+2s=OE+x=5. Therefore, x=2.

Since OE=5, the side length of the inscribed square is s=OE−2=5−2=3.

sandwich May 2, 2023