SAT Question a4c0547f: Right Triangles And Trigonometry answer and explanation with Desmos solution

Choice B is correct. It's given that angle $Z$ in triangle $XYZ$ is a right angle. Thus, side $YZ$ is the leg opposite angle $X$ and side $XZ$ is the leg adjacent to angle $X$. The tangent of an acute angle in a right triangle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. It follows that $\tan X=\frac{YZ}{XZ}$. It's given that $\tan X=\frac{12}{35}$ and the length of side $YZ$ is $24$ units. Substituting $\frac{12}{35}$ for $\tan X$ and $24$ for $YZ$ in the equation $\tan X=\frac{YZ}{XZ}$ yields $\frac{12}{35}=\frac{24}{XZ}$. Multiplying both sides of this equation by $35\left(XZ\right)$ yields $12\left(XZ\right)=24\left(35\right)$, or $12\left(XZ\right)=840$. Dividing both sides of this equation by $12$ yields $XZ=70$. The length $XY$ can be calculated using the Pythagorean theorem, which states that if a right triangle has legs with lengths of $a$ and $b$ and a hypotenuse with length $c$, then $a^2+b^2=c^2$. Substituting $70$ for $a$ and $24$ for $b$ in this equation yields $70^2+24^2=c^2$, or $5,476=c^2$. Taking the square root of both sides of this equation yields $\pm 74=c$. Since the length of the hypotenuse must be positive, $74=c$. Therefore, the length of $XY$ is $74$ units. The perimeter of a triangle is the sum of the lengths of all sides. Thus, $\left(74+70+24\right)$ units, or $168$ units, is the perimeter of triangle $XYZ$.