3420.25 – The Sum of Two Triangles

Sam and Dawn are equilateral triangles. Sam has side length 4; Dawn has side length 3. They are also brother and sister. Their mother, Simone, is also an equilateral triangle. Her area is the sum of the areas of Sam and Dawn. What is the length of Simone's side?

Solution

The area of an equilateral triangle having side $s$ is $\dfrac{s^2\sqrt3}{2}$ (you can check this).

Sam's area is $\dfrac{4^2\sqrt3}{2}$ , and Dawn's area is $\dfrac{3^2\sqrt3}{2}$ . Thus Simone's area is $\dfrac{5^2\sqrt3}{2}.$

The answer is $5$ .