Arcs and Chords
Overview
Chords and their arcs rise and fall together: in the same circle, congruent chords cut off congruent arcs, and congruent chords sit the same distance from the center. The most useful fact in the lesson is what happens when a diameter or radius meets a chord at a right angle — it cuts the chord into two congruent segments and cuts its arc in half as well, which often sets up a right triangle.
Vocabulary
- Chord
- A segment whose two endpoints lie on the circle.
- Arc of a Chord
- The minor arc whose endpoints are the endpoints of the chord.
- Congruent Chords
- Chords of the same circle (or of congruent circles) that have the same length.
- Equidistant from the Center
- Two chords are equidistant from the center when the perpendicular distances from the center to the chords are equal. Distance from a point to a line is always measured along a perpendicular.
- Bisector
- A line, ray, or segment that divides a segment or an arc into two congruent parts.
Main Ideas
- In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
- In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
- If a diameter or radius is perpendicular to a chord, then it bisects the chord and its arc.
- Because the perpendicular from the center bisects the chord, many chord problems become right-triangle problems: the hypotenuse is a radius, one leg is half the chord, and the other leg is the distance from the center.
Lesson Notes
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Lesson Video
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Arcs and Chords
Arcs and Chords
Practice Problems
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Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.