What is the exact length (in inches) of common internal tangent AB? (Hint: Use similar triangles to find OD and DP. Then apply the Pythagorean Theorem twice.)

Question

The center of a circle of radius 3 in. is at a distance of 40 in. from the center of a circle of radius 21 in.

What is the exact length (in inches) of common internal tangent AB?

 (Hint: Use similar triangles to find OD and DP. Then apply the Pythagorean Theorem twice.)

Answer 1

From similyarity of triangles OAD and PBD: 3/21 = OD/(40 - OD); 7OD = 40 - OD; OD = 5 in and PD = 35 in

By Pythagorian Theorem AD = √5² - 3² = 4; BD = 28. So, AB = 28 + 4 = 32 in