Circles, Chords, Diameters, and Their Relationships

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

Worksheets for Geometry

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 𝐴𝐵 below (as you did in Module 1). Draw another line that bisects 𝐴𝐵 but is not perpendicular to it. List one similarity and one difference between the two bisectors.

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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 angle’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 𝐴 is said to be equidistant from two different points 𝐵 and 𝐶 if 𝐴𝐵 = 𝐴𝐶

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