SAT Question f389569d: Right Triangles And Trigonometry answer and explanation with Desmos solution

Choice C is correct. It’s given that the equilateral triangle has a height of $39\sqrt{3}$. In equilateral triangles, all three angles have a measure of $60^{\circ }$ and the height is equal to the length of an altitude from one vertex to the midpoint of the side across from the vertex. Thus, an altitude in this equilateral triangle has a length of $39\sqrt{3}$. The altitude divides the equilateral triangle into two congruent 30-60-90 right triangles, where the altitude is the side across from the $60^{\circ }$ angle in each 30-60-90 right triangle. Since the altitude has a length of $39\sqrt{3}$, it follows from the properties of 30-60-90 right triangles that the side across from each $30^{\circ }$ angle has a length of $39$ and each hypotenuse has a length of $2(39)$, or $78$. Therefore, each side of the equilateral triangle has a length of $78$. In equilateral triangles, all three sides have equal length. Therefore, the perimeter of this equilateral triangle is $78+78+78$, or $234$.