Question

Let $C_1,$ $C_2,$ and $C_3$ be three parallel chords of a circle on the same side of the center. The distance between $C_1$ and $C_2$ is the same as the distance between $C_2$ and $C_3.$ The lengths of the chords are $20,$ $16,$ and $8.$ The radius of the circle is

$12$

$4\sqrt7$

$\frac{5\sqrt{65}}3$

$\frac{5\sqrt{22}}2$

not uniquely determined by the given information

Answer: D

Concepts:chord Pythagorean Theorem system of equations

Difficulty rating: 2100

Small Hint:

A perpendicular from the center bisects each chord

Big Hint:

Let the nearest chord be distance $u$ from the center and the common spacing be $v$

Solution:

Let $r$ be the radius, $u$ the distance to the $20$ -unit chord, and $v$ the common spacing. Then $\begin{aligned} r^2&=u^2+10^2,\\ r^2&=(u+v)^2+8^2,\\ r^2&=(u+2v)^2+4^2. \end{aligned}$ Consecutive subtraction gives $2uv+v^2=36$ and $2uv+3v^2=48,$ so $v^2=6$ and $u=\frac{15}{\sqrt6}.$ Hence $r^2=u^2+100=\frac{275}{2},$ and $r=\frac{5\sqrt{22}}{2}.$

Therefore, the correct answer is D.

1980 AMC 12 Problem 19

Answer: D

Difficulty rating: 2100

Small Hint:

Big Hint:

Solution:

Problem 19 in Other Years