teach me this ---**Question Stem:**
A circle has center G, and points M and N lie on the circle. Line segments MH and NH 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 GMHN is 3,856 millimeters, what is the distance, in millimeters, between points G and H?
**Options:**
A. 168
B. 1,752
C. 1,760
D. 1,768
视频信息
答案文本
视频字幕
Let's analyze this circle tangent problem step by step. We have a circle with center G and radius 168 millimeters. Points M and N are on the circle, and from external point H, we draw tangent segments MH and NH. The perimeter of quadrilateral GMHN is 3,856 millimeters, and we need to find the distance GH.
Now let's identify the key properties of tangent segments. First, tangent segments from an external point to a circle are equal in length, so MH equals NH. Second, the radius drawn to the point of tangency is perpendicular to the tangent line, creating right angles at M and N. We know the radius is 168 millimeters, and the perimeter of quadrilateral GMHN is 3,856 millimeters.
Now let's set up the perimeter equation. The perimeter of quadrilateral GMHN equals GM plus MH plus HN plus NG. Substituting our known values: 3856 equals 168 plus MH plus NH plus 168. Since tangent segments from an external point are equal, MH equals NH. This gives us 3856 equals 336 plus 2 times MH. Solving for MH: 2 times MH equals 3520, so MH equals 1760 millimeters.
Now we apply the Pythagorean theorem to the right triangle GMH. Since angle GMH is 90 degrees, we have GH squared equals GM squared plus MH squared. Substituting our values: GH squared equals 168 squared plus 1760 squared. Calculating: 168 squared is 28,224 and 1760 squared is 3,097,600. Adding these gives us GH squared equals 3,125,824. Taking the square root, we get GH equals 1768 millimeters.
Let's verify our answer by checking which option gives us GH squared equals 3,125,824. Option A: 168 squared equals 28,224. Option B: 1752 squared equals 3,069,504. Option C: 1760 squared equals 3,097,600. Option D: 1768 squared equals 3,125,824, which matches our calculation perfectly. Therefore, the distance between points G and H is 1768 millimeters. The answer is D.