Exercise 24 Page 563

Let's begin with recalling the $30^{\circ }\text{- }60^{\circ }\text{- }90^{\circ }$ Triangle Theorem. This theorem tells us that the length of the hypotenuse of this right triangle is $2$ times the length of the shorter leg $s,$ and the length of the longer leg is $\sqrt{3}$ times the length of the shorter leg.

In our exercise we are given that an equilateral triangle has an altitude length of $18$ feet and asked to determine the length of a side of this triangle. Notice that the altitude in a right triangle divides an equilateral triangle into two $30^{\circ }\text{- }60^{\circ }\text{- }90^{\circ }$ triangles.

Let $2s$ be the side length of the equilateral triangle and $s$ be half of this side length.

According to the theorem we recalled at the beginning, we can create an equation for $s.$ Let's recall that in $30^{\circ }\text{- }60^{\circ }\text{- }90^{\circ }$ triangles the longer leg is $\sqrt{3}$ times the length of the shorter leg. We can solve the above equation. Finally, in $30^{\circ }\text{- }60^{\circ }\text{- }90^{\circ }$ triangles the length of the hypotenuse is $\text{twice}$ the length of the shorter leg, so we can evaluate the side length of an equilateral triangle. The side length of this equilateral triangle is $12\sqrt{3}$ or approximately $20.79$ feet.