SAT Question 099526fc: Area and Volume answer and explanation with Desmos solution

Choice B is correct. The length of a segment in the xy-plane can be found using the distance formula, $\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$, where $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are the endpoints of the segment. The segment shown has endpoints at $\left(-6,4\right)$ and $\left(3,10\right)$. Substituting $\left(-6,4\right)$ and $\left(3,10\right)$ for $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, respectively, in the distance formula yields $\sqrt{{\left(3-\left(-6\right)\right)}^2+{\left(10-4\right)}^2}$, or $\sqrt{9^2+6^2}$, which is equivalent to $\sqrt{81+36}$, or $\sqrt{117}$. Let $x$ represent the length, in units, of the other leg of this triangle. The area, $A$, of a right triangle can be calculated using the formula $A=\frac{1}{2}bh$, where $b$ and $h$ are the lengths of the legs of the triangle. It's given that the area of the triangle is $36\sqrt{13}$ square units. Substituting $36\sqrt{13}$ for $A$, $\sqrt{117}$ for $b$, and $x$ for $h$ in the formula $A=\frac{1}{2}bh$ yields $36\sqrt{13}=\frac{1}{2}\left(\sqrt{117}\right)\left(x\right)$. Multiplying both sides of this equation by $2$ yields $72\sqrt{13}=x\sqrt{117}$. Dividing both sides of this equation by $\sqrt{117}$ yields $\frac{72\sqrt{13}}{\sqrt{117}}=x$. Multiplying the numerator and denominator on the left-hand side of this equation by $\sqrt{117}$ yields $\frac{72\sqrt{1,521}}{117}=x$, or $\frac{72\left(39\right)}{117}=x$, which is equivalent to $\frac{2,808}{117}=x$, or $24=x$. Therefore, the length, in units, of the other leg of this triangle is $24$.