NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY Lesson 2: Circles, Chords, Diameters, and Their Relationships Student Outcomes . Students identify the relationships between the diameters of a circle and other chords of the circle. Lesson Notes Students are asked to construct the perpendicular bisector of a line segment and draw conclusions about points on that bisector and the endpoints of the segment. They relate the construction to the theorem stating that any perpendicular bisector of a chord must pass through the center of the circle. Students should be made aware that figures are not drawn to scale. Classwork Opening Exercise (4 minutes) Scaffolding: Post a diagram and display the Opening Exercise steps to create a perpendicular Construct the perpendicular bisector of 푨푩̅̅̅̅ below (as you did in Module 1). bisector used in Lesson 4 of Module 1. Label the endpoints of the segment 퐴 and 퐵. Draw circle 퐴 with center ̅̅̅̅ Draw another line that bisects 푨푩̅̅̅̅ but is not perpendicular to it. 퐴 and radius 퐴퐵. List one similarity and one difference between the two bisectors. Draw circle 퐵 with center 퐵 and radius ̅퐵퐴̅̅̅. Answers will vary. Both bisectors divide the segment into two shorter segments of equal length. All points on the perpendicular bisector are equidistant from points 푨 and 푩. Points on the other . Label the points of bisector are not equidistant from points 푨 and 푩. The perpendicular bisector meets 푨푩̅̅̅̅ at right intersection as 퐶 and 퐷. angles. The other bisector meets at angles that are not congruent. Draw 퐶퐷⃡ . Recall for students the definition of equidistant. EQUIDISTANT: A point 퐴 is said to be equidistant from two different points 퐵 and 퐶 if 퐴퐵 = 퐴퐶. Points 퐵 and 퐶 can be replaced in the definition above with other figures (lines, etc.) as long as the distance to those figures is given meaning first. In this lesson, students define the distance from the center of a circle to a chord. This definition allows them to talk about the center of a circle as being equidistant from two chords. Lesson 2: Circles, Chords, Diameters, and Their Relationships 21 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY Discussion (12 minutes) Ask students independently or in groups to each draw chords and describe what they notice. Answers will vary depending on what each student drew. Lead students to relate the perpendicular bisector of a line segment to the points on a circle, guiding them toward seeing the relationship between the perpendicular bisector of a chord and the center of a circle. Construct a circle of any radius, and identify the center as point 푃. Draw a chord, and label it ̅퐴퐵̅̅̅. Construct the perpendicular bisector of ̅퐴퐵̅̅̅. What do you notice about the perpendicular bisector of ̅퐴퐵̅̅̅? It passes through point 푃, the center of the circle. Draw another chord, and label it ̅퐶퐷̅̅̅. Construct the perpendicular bisector of ̅퐶퐷̅̅̅. What do you notice about the perpendicular bisector of ̅퐶퐷̅̅̅? It passes through point 푃, the center of the circle. What can you say about the points on a circle in relation to the center of the circle? The center of the circle is equidistant from any two points on the circle. Look at the circles, chords, and perpendicular bisectors created by your neighbors. What statement can you MP.3 make about the perpendicular bisector of any chord of a circle? Why? & It must contain the center of the circle. The center of the circle is equidistant from the two endpoints of MP.7 the chord because they lie on the circle. Therefore, the center lies on the perpendicular bisector of the chord. That is, the perpendicular bisector contains the center. How does this relate to the definition of the perpendicular bisector of a line segment? Scaffolding: The set of all points equidistant from two given points (endpoints of a . Review the definition of line segment) is precisely the set of all points on the perpendicular central angle by posting a bisector of the line segment. visual guide. A central angle of a circle is an angle whose vertex is the center of a circle. 퐶 is the center of the circle below. 퐶 Lesson 2: Circles, Chords, Diameters, and Their Relationships 22 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY Exercises (20 minutes) Assign one proof to each group, and then jigsaw, share, and gallery walk as students present their work. Exercises 1. Prove the theorem: If a diameter of a circle bisects a chord, then it must be perpendicular to the chord. Draw a diagram similar to that shown below. Given: Circle 푪 with diameter 푫푬̅̅̅̅, chord 푨푩̅̅̅̅ , and 푨푭 = 푩푭 Prove: 푫푬̅̅̅̅ ⊥ 푨푩̅̅̅̅ 푨푭 = 푩푭 Given 푭푪 = 푭푪 Reflexive property 푨푪 = 푩푪 Radii of the same circle are equal in measure. △ 푨푭푪 ≅ △ 푩푭푪 SSS 풎∠푨푭푪 = 풎∠푩푭푪 Corresponding angles of congruent triangles are equal in measure. ∠푨푭푪 and ∠푩푭푪 are right angles Equal angles that form a linear pair each measure ퟗퟎ°. 푫푬̅̅̅̅ ⊥ 푨푩̅̅̅̅ Definition of perpendicular lines OR 푨푭 = 푩푭 Given 푨푪 = 푩푪 Radii of the same circle are equal in measure. 풎∠푭푨푪 = 풎∠푭푩푪 Base angles of an isosceles triangle are equal in measure. △ 푨푭푪 ≅ △ 푩푭푪 SAS 풎∠푨푭푪 = 풎∠푩푭푪 Corresponding angles of congruent triangles are equal in measure. ∠푨푭푪 and ∠푩푭푪 are right angles Equal angles that form a linear pair each measure ퟗퟎ°. 푫푬̅̅̅̅ ⊥ 푨푩̅̅̅̅ Definition of perpendicular lines Lesson 2: Circles, Chords, Diameters, and Their Relationships 23 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY 2. Prove the theorem: If a diameter of a circle is perpendicular to a chord, then it bisects the chord. Use a diagram similar to that in Exercise 1. Given: Circle 푪 with diameter 푫푬̅̅̅̅, chord 푨푩̅̅̅̅, and 푫푬̅̅̅̅ ⊥ 푨푩̅̅̅̅ Prove: 푫푬̅̅̅̅ bisects 푨푩̅̅̅̅ 푫푬̅̅̅̅ ⊥ 푨푩̅̅̅̅ Given ∠푨푭푪 and ∠푩푭푪 are right angles Definition of perpendicular lines △ 푨푭푪 and △ 푩푭푪 are right triangles Definition of right triangle ∠푨푭푪 ≅ ∠푩푭푪 All right angles are congruent. 푭푪 = 푭푪 Reflexive property 푨푪 = 푩푪 Radii of the same circle are equal in measure. △ 푨푭푪 ≅ △ 푩푭푪 HL 푨푭 = 푩푭 Corresponding sides of congruent triangles are equal in length. 푫푬̅̅̅̅ bisects 푨푩̅̅̅̅ Definition of segment bisector OR 푫푬̅̅̅̅ ⊥ 푨푩̅̅̅̅ Given ∠푨푭푪 and ∠푩푭푪 are right angles Definition of perpendicular lines ∠푨푭푪 ≅ ∠푩푭푪 All right angles are congruent. 푨푪 = 푩푪 Radii of the same circle are equal in measure. 풎∠푭푨푪 = 풎∠푭푩푪 Base angles of an isosceles triangle are congruent. 풎∠푨푪푭 = 풎∠푩푪푭 Two angles of triangle are equal in measure, so third angles are equal. △ 푨푭푪 ≅ △ 푩푭푪 ASA 푨푭 = 푩푭 Corresponding sides of congruent triangles are equal in length. 푫푬̅̅̅̅ bisects 푨푩̅̅̅̅ Definition of segment bisector Lesson 2: Circles, Chords, Diameters, and Their Relationships 24 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 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. Given: Circle 푶 with chords 푨푩̅̅̅̅ and 푪푫̅̅̅̅; 푨푩 = 푪푫; 푭 is the midpoint of 푨푩̅̅̅̅ and 푬 is the midpoint of 푪푫̅̅̅̅. Prove: 푶푭 = 푶푬 푨푩 = 푪푫 Given 푶푭̅̅̅̅ and 푶푬̅̅̅̅ are portions of diameters Definition of diameter 푶푭̅̅̅̅ ⊥ 푨푩̅̅̅̅; 푶푬̅̅̅̅ ⊥ 푪푫̅̅̅̅ If a diameter of a circle bisects a chord, then the diameter must be perpendicular to the chord. ∠푨푭푶 and ∠푫푬푶 are right angles Definition of perpendicular lines △ 푨푭푶 and △ 푫푬푶 are right triangles Definition of right triangle 푬 and 푭 are midpoints of 푪푫̅̅̅̅ and 푨푩̅̅̅̅ Given 푨푭 = 푫푬 푨푩 = 푪푫 and 푭 and 푬 are midpoints of 푨푩̅̅̅̅ and 푪푫̅̅̅̅. 푨푶 = 푫푶 All radii of a circle are equal in measure. △ 푨푭푶 ≅ △ 푫푬푶 HL 푶푬 = 푶푭 Corresponding sides of congruent triangles are equal in length. Lesson 2: Circles, Chords, Diameters, and Their Relationships 25 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 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.