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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

GeoB 6.4 Arcs Central Angles

  • 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.

GeoB 6.4 Chord Minor Arc

 

  • 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.

GeoB 6.4 Congruent Arcs

GeoB 6.4 Central Angles Arcs Cong

 


  • 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.

Geometric Proof Template

 

  • 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.

GeoB 6.4 Ex 01

 

  • 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°.

GeoB 6.4 Ex 02

 

  • 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.

GeoB 6.4 Ex 03

 

  • 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.

GeoB 6.4 Ex 04

 

  • 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°

GeoB 6.4 Ex 05

 


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