Problem 4
In the following diagram, the circle with center A has a radius of 6, and the circle with center B has a radius of 11. Line CD is tangent to both circles, at points C and D
respectively. Lines CD and AB intersect at point E, and the segment AE is 10. What is CD?
Solution
A line tangent to a circle is perpendicular to the radius of that circle, so two right triangles can be formed in the figure. We already know the length of two sides of
triangle AED, so we can use the Pythagorean Theorem to find the third.
(AD)^2 + (DE)^2 = (AE)^2
(6)^2 + (DE)^2 = (10)^2
(DE)^2 = 64
DE = 8
The two triangles are similar, since both are right triangles and they form a vertical angle, so all three angles of both triangles are the same. Therefore the ratios
of like sides will be equivilant.
CE/DE = CB/AD
CE/8 = 11/6
CE = 44/3 or approximately 14.67
CD = CE + DE
CD = 44/3 + 8
CD = 68/3 or approximately 22.67.
