1966 AMC 12 Problem 8

Question

The length of the common chord of two intersecting circles is

16

feet. If the radii are

10

feet and

17

feet, a possible value for the distance between the centers of the circles, expressed in feet, is:

27

21

\sqrt{389}

15

undetermined

The line of centers perpendicularly bisects the common chord

Find each center’s distance from the chord using half-chord

8

The half-chord has length

8.

The two perpendicular distances from the centers to its line are

\begin{gathered} \sqrt{10^2-8^2}=6,\\ \sqrt{17^2-8^2}=15. \end{gathered}

If the centers lie on opposite sides of the chord, their distance is

6+15=21,

which is a possible value.

Therefore, the correct answer is B.

Problem 8 in other years