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Arcs and Chords

Learn to recognize and apply relationships between arcs, chords, and diameters in circles. Prove and solve problems involving parallel lines, quadrilaterals, and circle properties. Use two-column proofs to demonstrate theorems.

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Arcs and Chords

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  1. Arcs and Chords Chapter 10-3

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

  3. Standard 7.0Students 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 21.0Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles. (Key) Lesson 3 CA

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

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

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

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

  8. Given: Prove: PROOFChoose the best reason to complete the following proof. Lesson 3 CYP1

  9. 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

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

  11. 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

  12. 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

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

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

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

  16. 10 Radius Perpendicular to a Chord A radius perpendicular to a chord bisects it. Def of seg bisector 8 Lesson 3 Ex3

  17. 10 8 Use the Pythagorean Theorem to find WJ. Pythagorean Theorem JK = 8, WK = 10 Simplify. Subtract 64 from each side. Take the square root of each side. Segment Addition Postulate WJ = 6, WL = 10 Subtract 6 from each side. 6 Lesson 3 Ex3

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

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

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

  21. 24 12 24 Chords Equidistant from Center Pythagorean Theorem 15 9 Lesson 3 Ex4

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

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

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

  25. Homework Chapter 10.3 • Pg 5749 – 31 all

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