3420.24 – Three Equilateral Triangles

Suppose that two equilateral triangles have side lengths of $x$ and $y$ . Find the side length of a third equilateral triangle whose area is the sum of the areas of the first two triangles.

Solution

The area of an equilateral triangle of side $s$ is $s^2 \times \dfrac{\sqrt{3}}{4}$ . Thus the areas of the first two triangles are $x^2 \times \dfrac{\sqrt{3}}{4}$ and $y^2 \times \dfrac{\sqrt{3}}{4}$ . The sum of these is

$\dfrac{(x^2 + y^2) \sqrt{3}}{4}$

so each side of the third triangle will be $\sqrt{x^2 + y^2}$ .