Properties of a Chord in a Circle

Perpendicular Bisector of a Chord and the Circle’s Center

1. The perpendicular bisector of a chord in a circle passes through the center of the circle.
2. A perpendicular line from the center of the circle to the chord bisects the chord.
   If , then
   
   

Proof:

1. In and , from the diagram:
   
   Therefore, by RHS (Right angle, Hypotenuse, Side) congruence, which implies:
   
   
2. In the next diagram, if you draw a chord in circle and place a point on the circle, then circle is the circumcenter of .
   
   
   Hence, the center is the intersection point of the perpendicular bisectors of the sides of , and the perpendicular bisector of passes through the center of the circle. In other words, the perpendicular bisector of a chord passes through the center of the circle.

Distance from the Center to the Chord and the Length of the Chord

In the same circle:

1. Chords that are equidistant from the center have equal lengths.
   If , then
2. Chords of equal length are equidistant from the center.
   If , then