Lesson 2 M5 GEOMETRY

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 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. This file derived from GEO-M5-TE-1.3.0 -10.2015

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

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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

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

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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

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

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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.

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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.

Given: Circle 푶 with chords 푨푩̅̅̅̅ and 푪푫̅̅̅̅; 푶푭 = 푶푬; 푭 is the midpoint of 푨푩̅̅̅̅ and 푬 is the midpoint of 푪푫̅̅̅̅.

Prove: 푨푩 = 푪푫

푶푭 = 푶푬 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 it 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

푨푶 = 푫푶 All radii of a circle are equal in measure.

△ 푨푭푶 ≅ △ 푫푬푶 HL

푨푭 = 푫푬 Corresponding sides of congruent triangles are equal in length.

푭 is the midpoint of 푨푩̅̅̅̅, and 푬 is the Given midpoint of 푪푫̅̅̅̅.

푨푩 = 푪푫 푨푭 = 푫푬 and 푭 and 푬 are midpoints of 푨푩̅̅̅̅ and 푪푫̅̅̅̅.

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 angle’s 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.

We are given that the two chords (푨푩̅̅̅̅ and 푪푫̅̅̅̅) are congruent. Since all radii of a circle are congruent, 푨푶̅̅̅̅ ≅ 푩푶̅̅̅̅̅ ≅ 푪푶̅̅̅̅ ≅ 푫푶̅̅̅̅̅. Therefore, △ 푨푩푶 ≅ △ 푫푪푶 by SSS. ∠푨푶푩 ≅ ∠푫푶푪 since corresponding angles of congruent triangles are equal in measure.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY

6. Prove the theorem: In a circle, if two chords define central angles equal in measure, then they are congruent.

Using the diagram from Exercise 5, we now are given that 풎∠푨푶푩 = 풎∠푪푶푫. Since all radii of a circle are congruent, 푨푶̅̅̅̅ ≅ 푩푶̅̅̅̅̅ ≅ 푪푶̅̅̅̅ ≅ 푫푶̅̅̅̅̅. Therefore, △ 푨푩푶 ≅ △ 푫푪푶 by SAS. 푨푩̅̅̅̅ ≅ 푫푪̅̅̅̅ because corresponding sides of congruent triangles are congruent.

Closing (4 minutes) Have students write all they know to be true about the diagrams below. Bring the class together, go through the Lesson Summary, having students complete the list that they started, and discuss each point. A reproducible version of the graphic organizer shown is included at the end of the lesson.

Diagram Explanation of Diagram Theorem or Relationship

If a diameter of a circle bisects a chord, then it must be perpendicular to the chord. Diameter of a circle bisecting a chord If a diameter of a circle is perpendicular to a chord, then it bisects the chord.

If two chords are congruent, then the center of a circle is equidistant from Two congruent chords equidistant the two chords. from center If the center of a circle is equidistant from two chords, then the two chords are congruent.

Congruent chords define central angles equal in measure. Congruent chords If two chords define central angles equal in measure, then they are congruent.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY

Lesson Summary

Theorems about chords and diameters in a circle and their converses:

. If a diameter of a circle bisects a chord, then it must be perpendicular to the chord. . If a diameter of a circle is perpendicular to a chord, then it bisects the chord. . If two chords are congruent, then the center is equidistant from the two chords. . If the center is equidistant from two chords, then the two chords are congruent. . Congruent chords define central angles equal in measure. . If two chords define central angles equal in measure, then they are congruent.

Relevant Vocabulary

EQUIDISTANT: A point 푨 is said to be equidistant from two different points 푩 and 푪 if 푨푩 = 푨푪.

Exit Ticket (5 minutes)

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY

Name Date

Lesson 2: Circles, Chords, Diameters, and Their Relationships

Exit Ticket

1. Given circle 퐴 shown, 퐴퐹 = 퐴퐺 and 퐵퐶 = 22. Find 퐷퐸.

2. In the figure, circle 푃 has a radius of 10. ̅퐴퐵̅̅̅ ⊥ ̅퐷퐸̅̅̅. a. If 퐴퐵 = 8, what is the length of 퐴퐶̅̅̅̅?

b. If 퐷퐶 = 2, what is the length of ̅퐴퐵̅̅̅?

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY

Exit Ticket Sample Solutions

1. Given circle 푨 shown, 푨푭 = 푨푮 and 푩푪 = ퟐퟐ. Find 푫푬.

ퟐퟐ

2. In the figure, circle 푷 has a radius of ퟏퟎ. 푨푩̅̅̅̅ ⊥ 푫푬̅̅̅̅. a. If 푨푩 = ퟖ, what is the length of 푨푪̅̅̅̅? ퟒ

b. If 푫푪 = ퟐ, what is the length of 푨푩̅̅̅̅?

ퟏퟐ

Problem Set Sample Solutions Students should be made aware that figures in the Problem Set are not drawn to scale.

1. In this drawing, 푨푩 = ퟑퟎ, 푶푴 = ퟐퟎ, and 푶푵 = ퟏퟖ. What is 푪푵?

√ퟑퟎퟏ ≈ ퟏퟕ. ퟑퟓ

2. In the figure to the right, 푨푪̅̅̅̅ ⊥ 푩푮̅̅̅̅, 푫푭̅̅̅̅ ⊥ 푬푮̅̅̅̅, and 푬푭 = ퟏퟐ. Find 푨푪.

ퟐퟒ

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY

3. In the figure, 푨푪 = ퟐퟒ, and 푫푮 = ퟏퟑ. Find 푬푮. Explain your work.

ퟓ, △ 푨푩푮 is a right triangle with 퐡퐲퐩퐨퐭퐞퐧퐮퐬퐞 = 퐫퐚퐝퐢퐮퐬 = ퟏퟑ and 푨푩 = ퟏퟐ, so 푩푮 = ퟓ by Pythagorean theorem. 푩푮 = 푮푬 = ퟓ.

4. In the figure, 푨푩 = ퟏퟎ, and 푨푪 = ퟏퟔ. Find 푫푬.

ퟒ

5. In the figure, 푪푭 = ퟖ, and the two concentric circles have radii of ퟏퟎ and ퟏퟕ. Find 푫푬.

ퟗ

6. In the figure, the two circles have equal radii and intersect at points 푩 and 푫. 푨 and 푪 are centers of the circles. 푨푪 = ퟖ, and the radius of each circle is ퟓ. 푩푫̅̅̅̅̅ ⊥ 푨푪̅̅̅̅. Find 푩푫. Explain your work.

ퟔ 푩푨 = 푩푪 = ퟓ (radii)

푨푮 = 푮푪 = ퟒ 푩푮 = ퟑ (Pythagorean theorem)

푩푫 = ퟔ

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY

7. In the figure, the two concentric circles have radii of ퟔ and ퟏퟒ. Chord 푩푭̅̅̅̅ of the larger circle intersects the smaller circle at 푪 and 푬. 푪푬 = ퟖ. 푨푫̅̅̅̅ ⊥ 푩푭̅̅̅̅. a. Find 푨푫.

ퟐ√ퟓ

b. Find 푩푭.

ퟖ√ퟏퟏ

8. In the figure, 푨 is the center of the circle, and 푪푩 = 푪푫. Prove that 푨푪̅̅̅̅ bisects ∠푩푪푫.

Let 푨푬̅̅̅̅ and 푨푭̅̅̅̅ be perpendiculars from 푨 to 푪푩̅̅̅̅ and 푪푫̅̅̅̅, respectively.

푪푩 = 푪푫 Given

푨푬 = 푨푭 If two chords are congruent, then the center is equidistant from the two chords.

푨푪 = 푨푪 Reflexive property

풎∠푪푬푨 = 풎∠푪푭푨 = ퟗퟎ° Definition of perpendicular

△ 푪푬푨 ≅△ 푪푭푨 HL

풎∠푬푪푨 = 풎∠푭푪푨 Corresponding angles of congruent triangles are equal in measure.

푨푪̅̅̅̅ bisects ∠푩푪푫 Definition of angle bisector

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.

We are given that the two chords (푨푩̅̅̅̅ and 푪푫̅̅̅̅) are congruent. Therefore, a rigid motion exists that carries 푨푩̅̅̅̅ to 푪푫̅̅̅̅. The same rotation that carries 푨푩̅̅̅̅ to 푪푫̅̅̅̅ also carries 푨푶̅̅̅̅ to 푪푶̅̅̅̅ and 푩푶̅̅̅̅̅ to 푫푶̅̅̅̅̅. The angle of rotation is the measure of ∠푨푶푪, and the rotation is clockwise.

b. Give a rotation proof of the converse: If two chords define central angles of the same measure, then they must be congruent.

Using the same diagram, we are given that ∠푨푶푩 ≅ ∠푪푶푫. Therefore, a rigid motion (a rotation) carries ∠푨푶푩 to ∠푪푶푫. This same rotation carries 푨푶̅̅̅̅ to 푪푶̅̅̅̅ and 푩푶̅̅̅̅̅ to 푫푶̅̅̅̅̅. The angle of rotation is the measure of ∠푨푶푪, and the rotation is clockwise.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY

Graphic Organizer on Circles

Diagram Explanation of Diagram Theorem or Relationship

Lesson 2: Circles, Chords, Diameters, and Their Relationships 33