Chord and Arc Measures Notes

Chord and Arc Measures

Chords of Circles Theorem

• Part 1: In the same or congruent circles, two minor arcs are congruent if their corresponding chords are congruent.

• If chord AB is congruent to chord BC, then arc AB is congruent to arc BC and vice versa.
  

• Part 2: If a diameter is perpendicular to a chord, it bisects that chord and its corresponding arc.
  

Example of Congruence

• For congruent chords (chord SR and chord ST), you can set up an equation for their arcs if they are congruent.
  

Other Parts of Theorem

• Part 3: If one chord is the bisector of another and they're perpendicular, the first chord is the diameter.

• Part 4: Two chords are congruent if and only if they are equidistant from the circle's center.
  

Application in Examples

• Use given lengths (e.g., length RS) and arc measures to find unknown values. If arc measures and chord lengths are congruent, their equations can be set equal to each other.

• Use algebra to solve for unknowns.
  

Finding Lengths

• Find lengths of chords or arcs based on congruencies established by previously discussed theorem parts.

• Use Pythagorean theorem for right triangles formed within the circle.

• Example for calculation:
    - If chord lengths and distances from the center are given, apply them to find unknown chord lengths and angles.
  

Conclusion

• Use properties of circular chords and arcs effectively to solve geometric problems.

• Employ algebraic methods for finding unknown quantities based on established congruences.