A circle has center G , and points M and N lie on the circle. Line segments M H and

Question

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 ?

Accepted Answer

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 = 2 x , and dividing both sides of this equation by 2 yields 1,760 = 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 = c 2 . Taking the square root of both sides of this equation yields &#177; 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 .

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