jmc

Question

Circles with centers A and B have radii 3 and 8, respectively. A common internal tangent touches the circles at C and D, as shown. Lines AB and CD intersect at E, and AE = 5. What is CD? FIGURE

Accepted Answer

Radii AC and BD are each perpendicular to CD. By the Pythagorean Theorem, CE = 5^2 - 3^2 = 4. Because ACE and BDE are similar, we have DECE = BDAC, so DE = CE BDAC = 4 83 = 323. Therefore CD = CE + DE = 4 + 323 = 443.