Survey
* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project
Download Lesson 2: Circles, Chords, Diameters, and Their Relationships
Document related concepts
no text concepts found
Transcript
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY Lesson 2: Circles, Chords, Diameters, and Their Relationships Classwork Opening Exercise Construct the perpendicular bisector of Μ Μ Μ Μ π΄π΅ below (as you did in Module 1). Draw another line that bisects Μ Μ Μ Μ π΄π΅ but is not perpendicular to it. List one similarity and one difference between the two bisectors. Lesson 2: Circles, Chords, Diameters, and Their Relationships This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.8 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY Exercises Figures are not drawn to scale. 1. Prove the theorem: If a diameter of a circle bisects a chord, then it must be perpendicular to the chord. 2. Prove the theorem: If a diameter of a circle is perpendicular to a chord, then it bisects the chord. Lesson 2: Circles, Chords, Diameters, and Their Relationships This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.9 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY 3. The distance from the center of a circle to a chord is defined as the length of the perpendicular segment from the center to the chord. Note that since this perpendicular segment may be extended to create a diameter of the circle, the segment also bisects the chord, as proved in Exercise 2. Prove the theorem: In a circle, if two chords are congruent, then the center is equidistant from the two chords. Use the diagram below. Lesson 2: Circles, Chords, Diameters, and Their Relationships This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.10 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY 4. Prove the theorem: In a circle, if the center is equidistant from two chords, then the two chords are congruent. Use the diagram below. Lesson 2: Circles, Chords, Diameters, and Their Relationships This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.11 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 2 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY 5. A central angle defined by a chord is an angle whose vertex is the center of the circle and whose rays intersect the circle. The points at which the angleβs rays intersect the circle form the endpoints of the chord defined by the central angle. Prove the theorem: In a circle, congruent chords define central angles equal in measure. Use the diagram below. 6. Prove the theorem: In a circle, if two chords define central angles equal in measure, then they are congruent. Lesson 2: Circles, Chords, Diameters, and Their Relationships This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.12 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 2 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY Lesson Summary Theorems about chords and diameters in a circle and their converses: ο§ If a diameter of a circle bisects a chord, then it must be perpendicular to the chord. ο§ If a diameter of a circle is perpendicular to a chord, then it bisects the chord. ο§ If two chords are congruent, then the center is equidistant from the two chords. ο§ If the center is equidistant from two chords, then the two chords are congruent. ο§ Congruent chords define central angles equal in measure. ο§ If two chords define central angles equal in measure, then they are congruent. Relevant Vocabulary EQUIDISTANT: A point π΄ is said to be equidistant from two different points π΅ and πΆ if π΄π΅ = π΄πΆ. Problem Set Figures are not drawn to scale. 1. In this drawing, π΄π΅ = 30, ππ = 20, and ππ = 18. What is πΆπ? 2. In the figure to the right, Μ Μ Μ Μ π΄πΆ β₯ Μ Μ Μ Μ π΅πΊ , Μ Μ Μ Μ π·πΉ β₯ Μ Μ Μ Μ πΈπΊ , and πΈπΉ = 12. Find π΄πΆ. Lesson 2: Circles, Chords, Diameters, and Their Relationships This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 2 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY 3. In the figure, π΄πΆ = 24, and π·πΊ = 13. Find πΈπΊ. Explain your work. 4. In the figure, π΄π΅ = 10, and π΄πΆ = 16. Find π·πΈ. 5. In the figure, πΆπΉ = 8, and the two concentric circles have radii of 10 and 17. Find π·πΈ. 6. In the figure, the two circles have equal radii and intersect at points π΅ and π·. π΄ and πΆ are centers of the circles. Μ Μ Μ Μ . Find π΅π·. Explain your work. π΄πΆ = 8, and the radius of each circle is 5. Μ Μ Μ Μ π΅π· β₯ π΄πΆ Lesson 2: Circles, Chords, Diameters, and Their Relationships This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.14 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 2 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY 7. In the figure, the two concentric circles have radii of 6 and 14. Chord Μ Μ Μ Μ π΅πΉ of the larger circle intersects the smaller Μ Μ Μ Μ β₯ Μ Μ Μ Μ circle at πΆ and πΈ. πΆπΈ = 8. π΄π· π΅πΉ . a. Find π΄π·. b. Find π΅πΉ. 8. Μ Μ Μ Μ bisects β π΅πΆπ·. In the figure, π΄ is the center of the circle, and πΆπ΅ = πΆπ·. Prove that π΄πΆ 9. In class, we proved: Congruent chords define central angles equal in measure. a. Give another proof of this theorem based on the properties of rotations. Use the figure from Exercise 5. b. Give a rotation proof of the converse: If two chords define central angles of the same measure, then they must be congruent. Lesson 2: Circles, Chords, Diameters, and Their Relationships This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.15 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Related documents