In the following exercise, the task is to find the value of m + n, derived from the two integers from the distance length from point A to point B. To find the distance between these 2 points, one could diagram a triangular figure consisting of three vertices at point A, point B, and the center of the encompassing circle. To lastly find the sum of the integers m and n, one could take the integers from the simplified equivalent fraction symbolizing the distance AB and summing those two integers to finally provide the answer to the following exercise.
Similar triangles will make this problem easier to solve. Instead of directly finding the length you can look for the length of one side instead of both. You draw the lines from the middle of the big circle to the middle of one of the small circles to that small circle’s letter. Same thing for the other small circle. You then draw a line from the middle of one small circle to the middle of the other. You can then solve.
We know that the radius of the smaller triangles is 5 for each. If we connected the two centers of the smaller circles we would get a line with the length 10 with the endpoints at the centers of the smaller circles(line Q). Two more radii can be placed that will connect both midpoints to both the letters. These are necessary radii because they will be used in the future. We also know that the radius of the big circle is 13. So we connect the midpoint of the big circle to the letters. That will also overlap two of the small circle radii. This will create a small triangle with line Q. The small triangle will have sides of 10, 8, and 8. Now we have line AB. Line AB is equal to m/n. Since these are similar triangles we can easily find the answers. AB is equal to 13/8 * 10(the other line) which is equal to 130/8 (since we have to make the 10 into 10/1) which is simplified to 65/4. That is our m/n. 65 is m while 4 is n. Now we add 65 and 4 to get 69 or the answer D.