# Circles, Chords, Diameters, and Their Relationships

### New York State Common Core Math Geometry, Module 5, Lesson 2

Worksheets for Geometry, Module 5, Lesson 2

Student Outcomes

• Identify the relationships between the diameters of a circle and other chords of the circle.

Circles, Chords, Diameters, and Their Relationships

Classwork

Opening Exercise

Construct the perpendicular bisector of π΄π΅ below (as you did in Module 1). Draw another line that bisects π΄π΅ but is not perpendicular to it. List one similarity and one difference between the two bisectors.

Exercises

Figures are not drawn to scale.

1. Prove the theorem: If a diameter of a circle bisects a chord, then it must be perpendicular to the chord.
2. Prove the theorem: If a diameter of a circle is perpendicular to a chord, then it bisects the chord
3. The distance from the center of a circle to a chord is defined as the length of the perpendicular segment from the center to the chord. Note that since this perpendicular segment may be extended to create a diameter of the circle, the segment also bisects the chord, as proved in Exercise 2.
Prove the theorem: In a circle, if two chords are congruent, then the center is equidistant from the two chords.
Use the diagram below.
4. Prove the theorem: In a circle, if the center is equidistant from two chords, then the two chords are congruent.
Use the diagram below.
5. A central angle defined by a chord is an angle whose vertex is the center of the circle and whose rays intersect the circle. The points at which the angleβs rays intersect the circle form the endpoints of the chord defined by the central angle.
Prove the theorem: In a circle, congruent chords define central angles equal in measure.
Use the diagram below.
6. Prove the theorem: In a circle, if two chords define central angles equal in measure, then they are congruent.

Lesson Summary

Theorems about chords and diameters in a circle and their converses:

• If a diameter of a circle bisects a chord, then it must be perpendicular to the chord.
• If a diameter of a circle is perpendicular to a chord, then it bisects the chord.
• If two chords are congruent, then the center is equidistant from the two chords.
• If the center is equidistant from two chords, then the two chords are congruent.
• Congruent chords define central angles equal in measure.
• If two chords define central angles equal in measure, then they are congruent.

Relevant Vocabulary

EQUIDISTANT: A point π΄ is said to be equidistant from two different points π΅ and πΆ if π΄π΅ = π΄πΆ

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