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.

Answer 2

The distance between the second chord from the center of the circle is 'a' (half the length of the first chord).

What is a chord?

A circle chord is a straight line segment with both endpoints on a circular arc.

As the length of a chord is equal to its distance to the center of the circle, therefore, the length of the chord and the distance to the center of the circle is 2a. Thus, In ΔAOB the length of the third side of the triangle can be written as,

$OA^2= OB^2+AB^2\\\\OA^2 =(2a)^2+a^2\\\\OA^2=5a^2\\\\OA = a\sqrt5$

Now, OA and OQ are the radii of the circle and the length of the radius of the circle is a√5 units.

Now, the length of the second chord in the same circle is twice as long as the first one, therefore, the length of the half chord is 2a. Therefore, in ΔAOB the length of the perpendicular side of the triangle can be written as,

$OQ^2=OP^2+PQ^2\\\\(a\sqrt5)^2= OP^2+(2a)^2\\\\5a^2-4a^2=OP^2\\\\OP = a$

Hence, the distance between the second chord from the center of the circle is 'a' (half the length of the first chord).

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