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Circle Theorem for Arcs and Chords

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High School Math based on the topics required for the Regents Exam conducted by NYSED.

Theorem on Circles and Chords
In a circle, a radius perpendicular to a chord bisects the chord.
In a circle, a radius that bisects a chord is perpendicular to the chord.
In a circle, the perpendicular bisector of a chord passes through the center of the circle.

The following diagram gives the perpendicular bisector theorem for the chords in a circle. Scroll down the page for more examples and solutions.

Perpendicular Bisector in a Circle

Theorem on Congruent Chords
In a circle, or congruent circles, congruent chords have congruent arcs. (converse) In a circle, or congruent circles, congruent arcs have congruent chords.

The following diagram gives the theorem for congruent chords. Scroll down the page for more examples and solutions.

Congruent Chords Theorem




Arcs and Chords
Radius drawn perpendicular to a chord bisects the chord and the arc Circles: Arcs and Chords Pt 1
In a circle, or congruent circles, congruent chords have congruent arcs. (converse) In a circle, or congruent circles, congruent arcs have congruent chords. A diameter that is perpendicular to a chord bisects the chord and its arc Circles: Arcs and Chords Pt 2
In a circle, or congruent circles, congruent chords have congruent arcs. (converse) In a circle, or congruent circles, congruent arcs have congruent chords. A diameter that is perpendicular to a chord bisects the chord and its arc Circles: Arcs and Chords Pt 3
In a circle, or congruent circles, congruent chords have congruent arcs. (converse) In a circle, or congruent circles, congruent arcs have congruent chords. A diameter that is perpendicular to a chord bisects the chord and its arc Arcs and Chords
If a diameter is perpendicular to a chord, then it bisects the chord and its arc.

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