SAT Question 4c95c7d4: Right Triangles And Trigonometry answer and explanation with Desmos solution

Choice C is correct. It's given that the logo is in the shape of a trapezoid that consists of three congruent equilateral triangles, and that the perimeter of the trapezoid is $20$ centimeters (cm). Since the perimeter of the trapezoid is the sum of the lengths of $5$ of the sides of the triangles, the length of each side of an equilateral triangle is $\frac{20}{5}=4$ centimeters. Dividing up one equilateral triangle into two right triangles yields a pair of congruent $30^{\circ }$-$60^{\circ }$-$90^{\circ }$ triangles. The shorter leg of each right triangle is half the length of the side of an equilateral triangle, or $2$ cm. Using the Pythagorean Theorem, $a^2+b^2=c^2$, the height of the equilateral triangle can be found. Substituting $a=2$ and $c=4$ and solving for $b$ yields $\sqrt{4^2-2^2}=\sqrt{12}=2\sqrt{3}$ cm. The area of one equilateral triangle is $\frac{1}{2}bh$, where $b=4$ and $h=2\sqrt{3}$. Therefore, the area of one equilateral triangle is $\frac{1}{2}\times 4\times (2\sqrt{3})=4\sqrt{3}$ square centimeters. The shaded area consists of two such triangles, so its area is $2\times 4\sqrt{3}=8\sqrt{3}$ square centimeters.