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Geometry and Trigonometry / Circles Difficulty: Hard

A circle has center $G$ , and points $M$ and $N$ lie on the circle. Line segments $M H$ and $N H$ are tangent to the circle at points $M$ and $N$ , respectively. If the radius of the circle is $168$ millimeters and the perimeter of quadrilateral $G M H N$ is $3,856$ millimeters, what is the distance, in millimeters, between points $G$ and $H$ ?

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Explanation

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 $G M$ and $G N$ are both radii. Therefore, segments $G M$ and $G N$ each have length $168$ millimeters. Two segments that are tangent to a circle and have a common exterior endpoint have equal length. Therefore, segment $M H$ and segment $N H$ have equal length. Let $x$ represent the length of segment $M H$ . Then $x$ also represents the length of segment $N H$ . It's given that the perimeter of quadrilateral $G M H N$ 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 $3,856 = 336 + 2 x$ . Subtracting $336$ from both sides of this equation yields $3,520 equals 2 x$ , and dividing both sides of this equation by $2$ yields $1,760 equals x$ . Therefore, the length of segment $M H$ 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 $G M$ is perpendicular to segment $M H$ . Since perpendicular segments form right angles, angle $G M H$ is a right angle. Therefore, triangle $G M H$ is a right triangle with legs of length $1,760$ millimeters and $168$ millimeters, and hypotenuse $G H$ . 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 $3,125,824 equals c squared$ . 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 $G H$ is $1,768$ millimeters, so the distance between points $G$ and $H$ is $1,768$ millimeters.

Choice A is incorrect. This is the distance between points $G$ and $M$ and between points $G$ and $N$ , not the distance between points $G$ and $H$ .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the distance between points $M$ and $H$ and between points $N$ and $H$ , not the distance between points $G$ and $H$ .