Question 41·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

Double the circle's radius to get its full width: $2(65)=130$. This matches the given length of $PQ$. A diameter goes all the way across the circle through its center, so $PQ$ is a diameter.

The corner of $\angle PRQ$ is at $R$ on the circle, and its sides reach the ends of diameter $PQ$. An inscribed angle spanning a diameter is half a $180^\circ$ semicircle: $\frac{180^\circ}{2}=90^\circ$. So the right angle is at $R$.

The diameter is the hypotenuse, the side across from the right angle. The Pythagorean theorem says the legs' squares add to the hypotenuse's square, so $PQ$ goes alone: $QR^2+126^2=130^2$. Here $126$ is $PR$, one leg, and $QR$ is the leg you need.

Type $x_1^2+126^2\sim130^2\left\{x_1>0\right\}$. The $\sim$ asks Desmos to fit $x_1$ in a regression, and the braces require a positive length. Under PARAMETERS, Desmos shows $x_1=32$, so $QR=32$.

The ratio $\frac{QR}{PR}$ means divide $QR$ by $PR$, so use $PR=126$ as the denominator. Type $x_1/126$ on the next Desmos line. It shows $0.253968\ldots$; tap its fraction button to see $\frac{16}{63}$. So $\frac{QR}{PR}=\frac{16}{63}$. Choice C.