Lesson 10-3 Arcs and Chords

Learning Objectives

• In a 40-minute period, students will be able to recognize and use relationships between arcs, chords, and diameters, achieving 80% accuracy on an exit slip.

Content Standards

• G.C.2: Identify and describe relationships among inscribed angles, radii, and chords.

• G.MG.3: Apply geometric methods to solve problems, such as designing objects under constraints or working with typographic grids based on ratios.

• Mathematical Practices: 4 (Model with mathematics) and 3 (Construct viable arguments and critique the reasoning of others).

Lesson Overview: 10-3 Arcs and Chords

Theorem 10.2

• Theorem Statement: In the same circle or congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

• If chord FG is congruent to chord HJ, then the measure of arc FG is equal to the measure of arc HJ (FG=HJ if and only if FG = HJ).

Proof of Theorem 10.2

1. Given: OP; QR = ST

2. Prove: QR ST

• Steps of proof include relationships of arcs and angles and the property of equality of radii.

• Conclusion states QR ST using CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Application of Concepts

Example: Finding Arc Measure Using Congruent Chords

• Scenario: Jewelry piece with congruent chords JM and KL, where the measure of arc KL is given as 90.

• Conclusion: mKL = mJM = 90.

Check Your Progress

• If RS and TV are congruent chords and mRS = 85, find mTV:

  • A. 42.5 B. 85 C. 127.5 D. 170.

Example: Finding Chord Lengths Using Congruent Arcs

1. Given: A = OB and WX = YZ.

2. Find the length of WX using congruency of segments.

3. Solve equation: 7x-2 = 5x+6 to find x and thus WX.

Additional Check Your Progress Example

• In the figure where G = OH and RT = LM, find the length of LM:

  • Options: A. 6 B. 8 C. 9 D. 13.

Theorems on Chords and Measurements

Theorem 10.3

• If a diameter (or radius) is perpendicular to a chord, it bisects the chord and its corresponding arc.

Theorem 10.4

• The perpendicular bisector of a chord is a diameter (or radius) of the circle.

Example of Applying Theorems

• Given radius EG is perpendicular to chord DF, find the measures using bisection.

• Conclusion: mDE = 75.

Real-World Application Example: Ceramic Tile

• Problem: Given diameter AB is 18 inches and chord EF is 8 inches, calculate segment CD using the Pythagorean theorem after determining relevant segments.

Example Summary

• Find missing lengths by applying the appropriate theorems and geometric principles to arrive at the correct measures.

Chords Equidistant from Center

Theorem 10.5

• Two chords are congruent if and only if they are equidistant from the center of the circle.

• Example: Chords FG and JH are congruent.

Applying the Distance Formula

1. If PQ = PR with the equation 4x – 3 = 2x + 3, simplify to find x and subsequently PQ.

• After solving, result yields: PQ = 9.

Practice Problems

Homework Assignment

• Complete problems on page 719 (7-23) to reinforce concepts learned.