6.4 – Chord and Arc Relationships

Review

• Circle Size
• A circle’s size depends on the length of its radius
• The larger the radius, the larger the circle, and vice versa
• Two circles are congruent if they have congruent radii (same length)

• Central Angles
• Are measured in degrees
• Are formed by two radii
• Produce minor arcs along the circle’s circumference

• Minor Arcs
• The measure of a minor arc equals the measure of the central angle that intercepts it.
• The radii from the minor arc’s two endpoints form the sides of the central angle.
• A segment connects the minor arc’s two endpoints for a chord.

• Chords
• A chord is a segment with both endpoints on a circle.
• If two chords are the same distance from the center of a circle, then they are congruent.

• Arcs
• An arc is part of a circle with two endpoints.

Notes

• Arc Length and Central Angle Measure
• Notice that the arcs in this video have different lengths (in units), yet their central angles have the same measure (in degrees).

• Congruent Arcs
• Both of the following rules MUST be true for arcs to be congruent:
1. They have the same measure.
2. They must belong to the same circle or congruent circles.

• Congruent Minor Arcs Have Congruent Chords
• If the associated chords of two arcs are congruent, then the arcs are congruent.
• Converse: If two arcs are congruent, then their associated chords are congruent.
• Here’s the proof.  Copy it down using the template below.

• Chords and Central Angles
• Two chords are congruent if and only if their associated central angles are congruent.
• Two central angles are congruent if and only if their associated chords are congruent.

Examples

• Ex 1. Given $\odot{O}$ below, the arcs $\widehat{GH}$ and $\widehat{JK}$ must be congruent.

• Ex 2. Given $\odot{O}$ below, if $\widehat{GH}$ and $\widehat{JK}$ are congruent, what is the measure of $\angle GOH$?
• Because central angles are congruent if their corresponding arcs are congruent, $\angle KOJ$=68°.

• Ex 3. Given $\odot{O}$ below, if $\widehat{GH}$ and $\widehat{HJ}$ are congruent, what is the measure of $\angle GOH$?
• Since a circle measures 360, subtract the given angle (60°) to see what’s left? You get 300°.
• Since both arcs are congruent, divide 300° by 2 to get 150° each.

• Ex 4. Given $\odot{O}$ below, if $\widehat{GH}$ and $\widehat{JK}$ are congruent, what is the measure of chord $\overline{JK}$?
• When two arcs are congruent, their associated chords are congruent, so $\overline{JK}=9$.

• Ex 5. For two arcs to be congruent, they must be in the same circle or congruent circles and they must have the same central angle measure.

• Ex 6. If chord $\overline{GH}$ is congruent to chord $\overline{HJ}$, what is the measure of $\angle HOJ$?
• Since the chords are congruent, the corresponding arcs are congruent.  The central angles of those arcs are the same measure as those arcs, so the answer is 116°