Question

All red triangles in the picture are equilateral. What is the ratio of the sides of the outer and inner squares?

Answer 1

Let the sides of the shaded triangles are a units. The sides of the unshaded triangles are a, a, \(4\frac{a}{2}\sqrt{3}\) units.

Side of the outer square = \(\sqrt{3}\frac{a}{2}+\sqrt{3}\frac{a}{2}+a\)

\(\sqrt{3}a+a=a(\sqrt{3}+1)\)

Side of the inner square = \(\sqrt{3a}+a-a(a+a)=\sqrt{3}a+a-a-a\)

\(\sqrt{3}a=a(\sqrt{3}-1)\)

Ratio of the sides = \(a(\sqrt{3}+1):a(\sqrt{3}-1)\)

\(\sqrt{3}+1:\sqrt{3}-1\)