AB and CD are two chords of a circle at distance x cm and 5x cm from the center. If AB = 2CD, find the length of CD in terms of x.

Answer 1

draw a rough sketch, a circle with two chords, center at the origin O

AB x above O,

CD 5x above O

AB=2CD

construct two right triangles with radius r as the hypotenuse of both

use the Pythagorean Theorem

r^2= CD^2 + x^2

r^2= (CD/2)^2 + 5^2

set them equal

CD^2 + x^2 = CD^2/4 + 25x^2

24x^2 = CD^2 -CD^2/4 = (3/4)CD^2

(4/3)24x^2 = CD^2

CD^2 = 32x^2

CD = sqr32x^2 = 4xsqr2

CD = 4xsqr2

CD = about 5.66x