Question

The length of a chord is equal to its distance to the center of the circle. A second chord in the same circle is twice as long as the first one. How far is the second chord from the center?

Answer 1

Answer:

The distance from the center of the circle to the longer chord is twice smaller than the distance from the center to the shorter chord.

Step-by-step explanation:

The length of the chord AB is the same as the distance OC from the center to the cord. Let OC=2x, then CA=x. By the Pythagorean theorem, the radius r of the circle is

$r^2=OC^2+AC^2,\\ \\r^2=(2x)^2+x^2=5x^2,\\ \\r=\sqrt{5}x.$

The length of the arc ED is 4x.

Consider right triangle EFO. In this triangle, EF=2x, EO=r, then the distance OF is

$OF^2=OE^2-EF^2,\\ \\OF^2=5x^2-(2x)^2=x^2,\\ \\OF=x.$

The distance from the center of the circle to the longer chord is twice smaller than the distance from the center to the shorter chord.