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6.2 – Chords

Key Terms

  • Chord – Any line segment whose endpoints are on the circle.
  • Diameter – A line segment that contains (passes through) the center of the circle and has endpoints on the circle.
    • The diameter is a chord, as it has two endpoints ON the edge of the circle.
    • This term also refers to the length of this line segment
    • The diameter of a circle is twice the length of the radius.

 


Review

  • Circumference – the distance around a circle
    • To find the circumference, use 2\pi r or \pi d

 

  • Perpendicular Bisector
    • A line, ray, or segment
    • Intersects a segment at its midpoint
    • Forms right angles
    • Divides a segment in half
    • Is the shortest path between a line segment and a point not on that segment

 


Notes

  • Chords
    • Watch the video and draw an circle with chord \overline{AB}.

 

  • Congruence and Equidistance
    • If two chords are the same distance from the center of a circle, then they must be congruent.
    • If two chords of a given circle are congruent, then they must be equidistant from the center of the circle.

 

  • Perpendicular Chords and Radii
    • The shortest path between a line and any point (not on the line) is a path that is perpendicular to the line.
    • If a radius of a circle bisects a chord, then it is perpendicular to that chord.
    • If a radius of a circle is perpendicular to a chord, then it bisects that chord.

GeoB 6.2 Perp Radius Chord

 

  • Conditional Statement Proof
    • Again, if a radius of a circle is perpendicular to a chord, then it bisects that chord.
    • Prove that the Radius \overline{OC} Bisects Chord \overline{AB}

 

  • Converse Statement Proof (true in this case)
    • Again, if a radius of a circle bisects a chord, then it is perpendicular to that chord.
    • Prove that the Radius \overline{OC} is Perpendicular to Chord \overline{AB}

 

  • Template for Proofs

Geometric Proof Template

 


  • Diameter
    • Another name for a chord that passes through the center of a circle is the diameter.
    • Changing the diameter of a circle changes the size of the circle because the diameter is two radii; and, the radius defines the size of the circle.

GeoB 6.2 Diameter Radius

 


Examples

  • Ex 1. In \odot{O}, radius \overline{OP} intersects chord \overline{AC} in point B so that AB = 8 units and BC = 8 units.
    • This means that \overline{OP} is perpendicular to \overline{AC}.
    • Try drawing this and marking congruency.

 

  • Ex 2. Given \odot{V} below, you can conclude that \overline{QS} is congruent to \overline{NR}

GeoB 6.2 Quiz1ex1

 

  • Ex 3. Given \odot{M} below, you can conclude that \overline{ML} is congruent to \overline{MA}

GeoB 6.2 Quiz1ex2

 

  • Ex 4. If the blue radius below is perpendicular to the green chord and the segment \overline{AB} is 9 units long, what is the length of the chord?
    • Since \overline{AC} has been bisected by the radius of the circle, \overline{AB} is half of \overline{AC}.
    • \overline{AB} and \overline{BC} are congruent, so \overline{BC} also equals 9.
    • 9 + 9 = 18, so \overline{AC}=18.

GeoB 6.2 Quiz2ex1

 

  • Ex 5. The blue segment below is a radius of \odot{O}. What is the length of the diameter of the circle?
    • Since the diameter is twice the length of the radius: 9.3(2) = 18.6 units is the length of the diameter.

GeoB 6.2 Quiz3ex1

 

  • Ex 6. The blue segment below is a diameter of \odot{O}. What is the length of the radius of the circle?
    • Since the diameter is twice the length of the radius: \frac{7.7}{2}=3.85 units is the length of the radius.

GeoB 6.2 Quiz3ex2

 


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