2021 AMC 12B Spring Problem 8

Question

Three equally spaced parallel lines intersect a circle, creating three chords of lengths

38,

38,

and

34.

What is the distance between two adjacent parallel lines?

5\tfrac{1}{2}

6

6\tfrac{1}{2}

7

7\tfrac{1}{2}

Equal chords are equidistant from the center, so the two

38

chords sit at heights

\pm\tfrac{d}{2}

With spacing

d,

use

r^2-\left(\tfrac{d}{2}\right)^2=19^2

and

r^2-\left(\tfrac{3d}{2}\right)^2=17^2

Place the center at height

0.

Two equal chords lie at equal distances from the center, so the three equally spaced lines are at heights

-\tfrac{d}{2},\tfrac{d}{2},\tfrac{3d}{2},

with the two

38

-chords at

\pm\tfrac{d}{2}

and the

34

-chord at

\tfrac{3d}{2}.

Half-chord relations give

r^2-\left(\tfrac{d}{2}\right)^2=19^2

and

r^2-\left(\tfrac{3d}{2}\right)^2=17^2.

Subtracting,

2d^2=19^2-17^2=72,

so

d^2=36

and

d=6.

Thus, the correct answer is B.

Problem 8 in other years