1. Geometry
11.4 Color Theory

2. 11.4 Chords
Objectives
Determine the relationships between a chord and a diameter of a circle
Determine the relationships between congruent chords and their minor arcs
Prove the Diameter-Chord Theorem
Prove the Equidistant Chord Theorem and the Converse
Prove the Congruent-Chord Arc Theorem and the Converse
Prove the Segment-Chord Theorem

3. 87 𝑜
180 𝑜 −87 𝑜 = 93 𝑜
93 𝑜
180 𝑜 −93 𝑜 = 87 𝑜

4. All angles are inscribed (On The Circle) 𝑚 𝑋𝑊 +𝑚 𝑋𝑆 +𝑚 𝑆𝑊 = 360 𝑜
The angle measures are half of the arc measure.
𝑚∠𝑆+𝑚∠𝑊+𝑚∠𝑋= 𝑜 = 180 𝑜
All 3 angles also make a triangle.

5. Problem 1: Chords and Diameters
Discuss #1 and #2
Sketchpad
Perpendicular Bisector through Center
1g?
The perpendicular bisector of a chord always passes through the center of the circle

6. Problem 1: Chords and Diameters
Diameter-Chord Theorem
If a circle’s diameter is perpendicular to a chord, then the diameter bisects the chord and bisects the arc determined by the chord.
Reminder: The distance from a point to a segment is the perpendicular segment from the point to the segment.

7. Problem 1: Chords and Diameters
Draw #3
Sketchpad: Equidistant Chords

8. Problem 1: Chords and Diameters
Equidistant Chord Theorem
If two chords of the same circle or congruent circles are congruent, then they are equidistant from the center of the circle.
Equidistant Chord Converse Theorem
If two chords of the same circle or congruent circles are equidistant from the center of the circle, then the chords are congruent.

9. Problem3: Chords and Arcs
Sketchpad: Congruent Chord Congruent Arc
Congruent Chord-Congruent Arc Theorem
If two chords of the same circle or congruent circles are congruent, then their corresponding arcs are congruent.
Congruent Chord-Congruent Arc Converse Theorem
If two arcs of the same circle or congruent circles are congruent, then their corresponding chords are congruent.

10. Problem 4: Segments on Chords
Segments of a Chord
Segments formed on a chord when two chords of a circle intersect
Segments of Chord 𝑌𝑅
𝐸𝑌 𝑎𝑛𝑑 𝐸𝑅
Segments of Chord 𝑃𝑂
𝐸𝑃 𝑎𝑛𝑑 𝐸𝑂

11. Problem 4: Segments on Chords
Segment-Chord Theorem
If two chords in a circle intersect, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the second chord.
Sketchpad Demo

12. Collaborate: 2 Minutes
12∙8=10𝑥
96=10𝑥
𝑥=9.6 𝑓𝑒𝑒𝑡

13. Collaborate: 2 Minutes 𝐵𝐹 || 𝐶𝐸 𝐶𝐻 ≅ 𝐸𝐻 and 𝐶𝐷 ≅ 𝐸𝐷
Diameter-Chord Theorem
If a circle’s diameter is perpendicular to a chord, then the diameter bisects the chord and bisects the arc determined by the chord.

14. Formative Assessment Skills Practice 11.4
Match the Vocabulary Pg. 809 (1-7)
Pg (1-24)