# Congruent Chords in the Same Circle are Equidistant from the Center

Two congruent chords of a circle are equidistant from its center.

**NOTE:** Two chords are said to be congruent if they are equal in length.

Consider the equation of the circle

Suppose that and are congruent chords with , , and as shown in the given diagram. Since the circle passes through the points and , the equation of the circle becomes

Since is the midpoint of the chord , so

Since is the midpoint of the chord , so

Now we shall find the distance between and , as follows:

Similarly, we can show that

Since and are congruent chords,

Using equations (2) and (3), we get the following result:

Adding to both sides of the above equation, we have

Using equations (vi) and (vii) in equation (viii), we get

This shows that the congruent chords of a circle are equidistant from its center.