Arcs and Chords Chapter 10-3.

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Presentation transcript:

Arcs and Chords Chapter 10-3

Recognize and use relationships between arcs and chords. Recognize and use relationships between chords and diameters. inscribed circumscribed Lesson 3 MI/Vocab

Standard 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. (Key) Standard 21.0 Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles. (Key) Lesson 3 CA

Chord Theorems In the same circle or  circles, 2 minor arcs are   their corresponding chords are  B A C D E BC  ED  BC  ED

PROOF Write a two-column proof. Prove Theorem 10.2 PROOF Write a two-column proof. Prove: Given: is a semicircle. Lesson 3 Ex1

3. In a circle, if 2 chords are , corr. minor arcs are . 3. Prove Theorem 10.2 Answer: Proof: Statements Reasons 1. 1. Given is a semicircle. 2. Def. of semicircle 2. 3. In a circle, if 2 chords are , corr. minor arcs are . 3. 4. Def. of arcs 4. 5. Def. of arc measure 5. Lesson 3 Ex1

6. Arc Addition Postulate Prove Theorem 10.2 Answer: Statements Reasons 6. 6. Arc Addition Postulate 7. 7. Substitution 8. 8. Subtraction Property and simplify 9. 9. Division Property 10. 10. Def. of arc measure 11. 11. Substitution Lesson 3 Ex1

PROOF Choose the best reason to complete the following proof. Given: Prove: Lesson 3 CYP1

2. In a circle, 2 minor arcs are , chords are . Proof: Statements Reasons 1. 2. 3. 4. 1. Given 2. In a circle, 2 minor arcs are , chords are . 3. ______ 4. In a circle, 2 chords are , minor arcs are . Lesson 3 CYP1

A. Segment Addition Postulate B. Definition of  C. Definition of Chord D. Transitive Property A B C D Lesson 3 CYP1

Inscribed Polygons If all the vertices of a polygon lie on the circle The polygon is inscribed in the circle The circle is circumscribed about the polygon

A regular hexagon is inscribed in a circle as part of a logo for an advertisement. If opposite vertices are connected by line segments, what is the measure of angle P in degrees? Since connecting the opposite vertices of a regular hexagon divides the hexagon into six congruent triangles, each central angle will be congruent. The measure of each angle is 360 ÷ 6 or 60. Answer: 60 Lesson 3 Ex2

ADVERTISING A logo for an advertising campaign is a pentagon that has five congruent central angles. Determine whether A. yes B. no C. cannot be determined A B C Lesson 3 CYP2

Chord Theorems If the diameter of a circle is  to a chord,  the diameter bisects the chord and its arc A B C D AD  DC AB  BC

Radius Perpendicular to a Chord Since radius is perpendicular to chord Arc addition Substitution Substitution Subtraction Lesson 3 Ex3

Radius Perpendicular to a Chord A radius perpendicular to a chord bisects it. Def of seg bisector 10 8 Lesson 3 Ex3

Use the Pythagorean Theorem to find WJ. JK = 8, WK = 10 Simplify. Subtract 64 from each side. Take the square root of each side. 8 10 Segment Addition Postulate WJ = 6, WL = 10 Subtract 6 from each side. 6 Lesson 3 Ex3

A. 35 B. 70 C. 105 D. 145 A B C D Lesson 3 CYP3

A. 15 B. 5 C. 10 D. 25 A B C D Lesson 3 CYP3

Chord Theorems In the same circle or  circles, 2 chords are   they are equidistant from the center. EF  EG  AB  CD & AB  CD F G E C D A B

Chords Equidistant from Center 24 Pythagorean Theorem 15 12 9 24 Lesson 3 Ex4

A. 12 B. 36 C. 72 D. 32 A B C D Lesson 3 CYP4

A. 12 B. 36 C. 72 D. 32 A B C D Lesson 3 CYP4

Chord Theorems Sample Problem AD  DC Solve for x + y 3x + 7 = 5x + 3 4 = 2x 2=x AD = 3x + 7; DC = 5x +3 m AB = 4y + 8; m AEC = 96 A C D E AB  BC AB  ½ AC m AC = m AEC m AC = 96 4y + 8 = ½ (96) 4y + 8 = 48 4y = 40 y = 10 B

Homework Chapter 10.3 Pg 574 9 – 31 all