SAT Question 0acfddb5: Circles answer and explanation with Desmos solution

Choice D is correct. It's given that the radius of the circle is $168$ millimeters. Since points $M$ and $N$ both lie on the circle, segments $GM$ and $GN$ are both radii. Therefore, segments $GM$ and $GN$ each have length $168$ millimeters. Two segments that are tangent to a circle and have a common exterior endpoint have equal length. Therefore, segment $MH$ and segment $NH$ have equal length. Let $x$ represent the length of segment $MH$. Then $x$ also represents the length of segment $NH$. It's given that the perimeter of quadrilateral $GMHN$ is $3,856$ millimeters. Since the perimeter of a quadrilateral is equal to the sum of the lengths of the sides of the quadrilateral, $3,856=168+168+x+x$, or $3,856=336+2x$. Subtracting $336$ from both sides of this equation yields $3,520=2x$, and dividing both sides of this equation by $2$ yields $1,760=x$. Therefore, the length of segment $MH$ is $1,760$ millimeters. A line segment that's tangent to a circle is perpendicular to the radius of the circle at the point of tangency. Therefore, segment $GM$ is perpendicular to segment $MH$. Since perpendicular segments form right angles, angle $GMH$ is a right angle. Therefore, triangle $GMH$ is a right triangle with legs of length $1,760$ millimeters and $168$ millimeters, and hypotenuse $GH$. By the Pythagorean theorem, if a right triangle has a hypotenuse with length $c$ and legs with lengths $a$ and $b$, then $a^2+b^2=c^2$. Substituting $1,760$ for $a$ and $168$ for $b$ in this equation yields $1,760^2+168^2=c^2$, or $3,125,824=c^2$. Taking the square root of both sides of this equation yields $\pm 1,768=c$. Since $c$ represents a length, which must be positive, the value of $c$ is $1,768$. Therefore, the length of segment $GH$ is $1,768$ millimeters, so the distance between points $G$ and $H$ is $1,768$ millimeters.