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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5

Lesson 2: , Chords, , and Their Relationships

Student Outcomes . Identify the relationships between the diameters of a and other chords of the circle.

Lesson Notes Students are asked to construct the bisector of a 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 must pass through the center of the circle.

Classwork Opening Exercise (4 minutes) Scaffolding: Post a diagram and display the steps to create a Opening Exercise perpendicular bisector used in Construct the perpendicular bisector of below (as you did in Module 1). Lesson 4 of Module 1. . 𝑨𝑨𝑨𝑨���� Label the endpoints of the segment and . . Draw circle with center 𝐴𝐴 𝐵𝐵 and . 𝐴𝐴 Draw another line that bisects but is not perpendicular to it. . Draw circle with center 𝐴𝐴 �𝐴𝐴��𝐴𝐴� and radius . List one and one difference𝑨𝑨𝑨𝑨���� between the two bisectors. 𝐵𝐵 . Label the points of Answers will vary. All points on the perpendicular bisector are equidistant from points and 46T . 𝐵𝐵 �𝐵𝐵��𝐵𝐵� Points on the other bisector are not equidistant from points A and B. The perpendicular bisector intersection as and . 𝑨𝑨 𝑩𝑩 meets at right . The other bisector meets at angles that are not congruent. . Draw . 𝐶𝐶 𝐷𝐷 𝑨𝑨𝑨𝑨���� 𝐶𝐶⃖���𝐶𝐶�⃗ You may wish to recall for students the definition of equidistant:

EQUIDISTANT:. A is said to be equidistant from two different points and if = .

Points and can be𝐴𝐴 replaced in the definition above with other figures (lines,𝐵𝐵 etc.)𝐶𝐶 𝐴𝐴as𝐴𝐴 long𝐴𝐴 as𝐴𝐴 the distance to those figures is given meaning first. In this lesson, we will define the distance from the center of a circle to a chord. This definition𝐵𝐵 will𝐶𝐶 allow us to talk about the center of a circle as being equidistant from two chords.

Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14 21

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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 . . Draw a chord, and label it . 𝑃𝑃 . Construct the perpendicular bisector of . �𝐴𝐴��𝐴𝐴� . What do you notice about the perpendicular bisector of ? �𝐴𝐴��𝐴𝐴�  It passes through point P, the center of the circle. �𝐴𝐴��𝐴𝐴� . Draw another chord and label it . . Construct the perpendicular bisector of . �𝐶𝐶��𝐶𝐶� . What do you notice about the perpendicular bisector of ? �𝐶𝐶��𝐶𝐶�  It passes through point , the center of the circle. �𝐶𝐶��𝐶𝐶� . What can you say about the points on a circle in relation to the center of the circle? 𝑃𝑃  The center of the circle is equidistant from any two points on the circle. MP.3 . Look at the circles, chords, and perpendicular bisectors created by your neighbors. What statement can you & make about the perpendicular bisector of any chord of a circle? Why? MP.7  It must contain the center of the circle. The center of the circle is equidistant from the two endpoints of 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 by posting a bisector of the line segment. visual guide. . A central angle of a circle

is an angle whose is the center of a circle.

Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14 22

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY

Exercises 1–6 (20 minutes) Assign one proof to each group, and then jigsaw, share, and gallery walk as students present their work.

Exercises 1–6

1. Prove the theorem: If a of a circle bisects a chord, then it must be perpendicular to the chord. Draw a diagram similar to that shown below.

Given: Circle with diameter , chord , and = .

Prove: 𝑪𝑪 �𝑫𝑫𝑫𝑫��� �𝑨𝑨��𝑩𝑩� 𝑨𝑨𝑨𝑨 𝑩𝑩𝑩𝑩

= �𝑫𝑫𝑫𝑫�� � ⊥ 𝑨𝑨𝑨𝑨���� Given

𝑨𝑨𝑨𝑨 = 𝑩𝑩𝑩𝑩 Reflexive property

𝑭𝑭𝑭𝑭 = 𝑭𝑭𝑭𝑭 Radii of the same circle are equal in measure

𝑨𝑨𝑨𝑨 𝑩𝑩𝑩𝑩 SSS

△ 𝑨𝑨𝑨𝑨𝑨𝑨 ≅=△ 𝑩𝑩𝑩𝑩𝑩𝑩 Corresponding angles of congruent are equal in measure

𝒎𝒎∠𝑨𝑨𝑨𝑨 and𝑨𝑨 𝒎𝒎∠𝑩𝑩𝑩𝑩𝑩𝑩 are right angles Equal angles that form a linear pair each measure °

∠𝑨𝑨𝑨𝑨𝑨𝑨 ∠𝑩𝑩𝑩𝑩𝑩𝑩 Definition of perpendicular lines 𝟗𝟗𝟗𝟗

𝑫𝑫𝑫𝑫OR���� ⊥ 𝑨𝑨𝑨𝑨���� = Given

𝑨𝑨𝑨𝑨 = 𝑩𝑩𝑩𝑩 Radii of the same circle are equal in measure

𝑨𝑨𝑨𝑨 𝑩𝑩𝑩𝑩= Base angles of an isosceles are equal in measure

𝒎𝒎∠𝑭𝑭𝑭𝑭𝑭𝑭 𝒎𝒎∠𝑭𝑭𝑭𝑭𝑭𝑭 SAS

△ 𝑨𝑨𝑨𝑨𝑨𝑨 ≅=△ 𝑩𝑩𝑩𝑩𝑩𝑩 Corresponding angles of congruent triangles are equal in measure

𝒎𝒎∠𝑨𝑨𝑨𝑨 and𝑨𝑨 𝒎𝒎∠𝑩𝑩𝑩𝑩𝑩𝑩 are right angles Equal angles that form a linear pair each measure °

∠𝑨𝑨𝑨𝑨𝑨𝑨 ∠𝑩𝑩𝑩𝑩𝑩𝑩 Definition of perpendicular lines 𝟗𝟗𝟗𝟗

𝑫𝑫𝑫𝑫���� ⊥ 𝑨𝑨𝑨𝑨����

Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14 23

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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 above.

Given: Circle with diameter , chord , and

Prove: bisects𝑪𝑪 �𝑫𝑫𝑫𝑫��� �𝑨𝑨𝑨𝑨��� �𝑫𝑫𝑫𝑫��� ⊥ 𝑨𝑨𝑨𝑨����

�𝑫𝑫𝑫𝑫�� � �𝑨𝑨𝑨𝑨��� Given

𝑫𝑫���𝑬𝑬� ⊥ 𝑨𝑨𝑨𝑨�and��� are right angles Definition of perpendicular lines

∠𝑨𝑨𝑨𝑨𝑨𝑨 and∠ 𝑩𝑩𝑩𝑩𝑩𝑩 are right triangles Definition of right

△ 𝑨𝑨𝑨𝑨𝑨𝑨 △ 𝑩𝑩𝑩𝑩𝑩𝑩 All right angles are congruent

∠𝑨𝑨𝑨𝑨=𝑨𝑨 ≅ ∠ 𝑩𝑩𝑩𝑩𝑩𝑩 Reflexive property

𝑭𝑭𝑭𝑭 = 𝑭𝑭𝑭𝑭 Radii of the same circle are equal in measure

𝑨𝑨𝑨𝑨 𝑩𝑩𝑩𝑩 HL

△ 𝑨𝑨𝑨𝑨=𝑨𝑨 ≅ △ 𝑩𝑩𝑩𝑩𝑩𝑩 Corresponding sides of congruent triangles are equal in

𝑨𝑨𝑨𝑨 bisects𝑩𝑩𝑩𝑩 Definition of segment bisector

𝑫𝑫𝑫𝑫OR���� �𝑨𝑨𝑨𝑨��� Given

𝑫𝑫𝑫𝑫���� ⊥ 𝑨𝑨𝑨𝑨�and��� are right angles Definition of perpendicular lines

∠𝑨𝑨𝑨𝑨𝑨𝑨 ∠𝑩𝑩𝑩𝑩𝑩𝑩 All right angles are congruent

∠𝑨𝑨𝑨𝑨=𝑨𝑨 ≅ ∠ 𝑩𝑩𝑩𝑩𝑩𝑩 Radii of the same circle are equal in measure

𝑨𝑨𝑨𝑨 𝑩𝑩𝑩𝑩= Base angles of an isosceles triangle are congruent

𝒎𝒎∠𝑭𝑭𝑭𝑭𝑭𝑭 = 𝒎𝒎∠𝑭𝑭𝑭𝑭𝑭𝑭 Two angles of triangle are equal in measure, so third angles are equal

𝒎𝒎∠𝑨𝑨𝑨𝑨𝑨𝑨 𝒎𝒎∠𝑩𝑩𝑩𝑩𝑩𝑩 ASA

△ 𝑨𝑨𝑨𝑨=𝑨𝑨 ≅△ 𝑩𝑩𝑩𝑩𝑩𝑩 Corresponding sides of congruent triangles are equal in length

𝑨𝑨𝑨𝑨 bisects𝑩𝑩𝑩𝑩 Definition of segment bisector

𝑫𝑫𝑫𝑫 ���� �𝑨𝑨𝑨𝑨���

Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14 24

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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. that, since this perpendicular segment may be extended to create a diameter of the circle, therefore, the segment also bisects the chord, as proved in Exercise 2 above. 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 with chords and ; = ; is the of and is the midpoint of .

Prove: =𝑶𝑶 𝑨𝑨𝑨𝑨���� �𝑪𝑪��𝑪𝑪� 𝑨𝑨𝑨𝑨 𝑪𝑪𝑪𝑪 𝑭𝑭 �𝑨𝑨𝑨𝑨��� 𝑬𝑬 𝑪𝑪𝑪𝑪

= 𝑶𝑶𝑶𝑶 𝑶𝑶𝑶𝑶 Given

𝑨𝑨𝑨𝑨 𝑪𝑪𝑪𝑪; If a diameter of a circle bisects a chord, then the diameter must be perpendicular to the chord. 𝑶𝑶���𝑶𝑶� ⊥ 𝑨𝑨𝑨𝑨���� �𝑶𝑶��𝑶𝑶� ⊥ 𝑪𝑪��𝑪𝑪�� and are right angles Definition of perpendicular lines

∠𝑨𝑨𝑨𝑨𝑨𝑨 and∠ 𝑫𝑫𝑫𝑫𝑫𝑫 are right triangles Definition of

△E and𝑨𝑨𝑨𝑨𝑨𝑨 are △ 𝑫𝑫𝑫𝑫𝑫𝑫 of and Given

= 𝑭𝑭 𝑪𝑪���𝑪𝑪� �𝑨𝑨𝑨𝑨��� = and and are midpoints of and

𝑨𝑨𝑨𝑨 = 𝑫𝑫𝑫𝑫 𝑨𝑨𝑨𝑨All radii𝑪𝑪 𝑪𝑪of a circle𝑭𝑭 are𝑬𝑬 equal in measure �𝑨𝑨𝑨𝑨��� �𝑪𝑪��𝑪𝑪�

𝑨𝑨𝑨𝑨 𝑫𝑫𝑫𝑫 HL

△ 𝑨𝑨𝑨𝑨=𝑨𝑨 ≅ △ 𝑫𝑫𝑫𝑫𝑫𝑫 Corresponding sides of congruent triangles are equal in length

𝑶𝑶𝑶𝑶 𝑶𝑶𝑶𝑶

Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14 25

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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 with chords and ; = ; is the midpoint of and is the midpoint of

Prove: =𝑶𝑶 𝑨𝑨𝑨𝑨���� �𝑪𝑪��𝑪𝑪� 𝑶𝑶𝑶𝑶 𝑶𝑶𝑶𝑶 𝑭𝑭 �𝑨𝑨𝑨𝑨��� 𝑬𝑬 𝑪𝑪𝑪𝑪

= 𝑨𝑨𝑨𝑨 𝑪𝑪𝑪𝑪 Given

𝑶𝑶𝑶𝑶 𝑶𝑶𝑶𝑶; If a diameter of a circle bisects a chord, then it must be perpendicular to the chord. 𝑶𝑶���𝑶𝑶� ⊥ 𝑨𝑨𝑨𝑨���� �𝑶𝑶��𝑶𝑶� ⊥ 𝑪𝑪��𝑪𝑪�� and are right angles Definition of perpendicular lines.

∠𝑨𝑨𝑨𝑨𝑨𝑨 and∠ 𝑫𝑫𝑫𝑫𝑫𝑫 are right triangles Definition of right triangle.

△ 𝑨𝑨𝑨𝑨=𝑨𝑨 △ 𝑫𝑫𝑫𝑫𝑫𝑫 All radii of a circle are equal in measure.

𝑨𝑨𝑨𝑨 𝑫𝑫𝑫𝑫 HL

△ 𝑨𝑨𝑨𝑨=𝑨𝑨 ≅ △ 𝑫𝑫𝑫𝑫𝑫𝑫 Corresponding sides of congruent triangles are equal in length.

𝑨𝑨𝑨𝑨 is the𝑫𝑫𝑫𝑫 midpoint of , and is the midpoint of Given 𝑭𝑭 �𝑨𝑨𝑨𝑨��� 𝑬𝑬 = 𝑪𝑪𝑪𝑪 = and and are midpoints of and .

𝑨𝑨𝑨𝑨 𝑪𝑪𝑪𝑪 𝑨𝑨𝑨𝑨 𝑫𝑫𝑫𝑫 𝑭𝑭 𝑬𝑬 �𝑨𝑨𝑨𝑨��� �𝑪𝑪��𝑪𝑪� 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 ( and ) are congruent. Since all radii of a circle are congruent, . Therefore, by SSS. since corresponding angles of congruent ���� ���� ���� triangles are equal in measure. 𝑨𝑨𝑨𝑨 𝑪𝑪𝑪𝑪 𝑨𝑨𝑨𝑨 ≅ 𝑩𝑩𝑩𝑩 ≅ 𝑪𝑪���𝑪𝑪� ≅ �𝑫𝑫��𝑫𝑫�� △ 𝑨𝑨𝑨𝑨𝑨𝑨 ≅ △ 𝑪𝑪𝑪𝑪𝑪𝑪 ∠𝑨𝑨𝑨𝑨𝑨𝑨 ≅ ∠𝑪𝑪𝑪𝑪𝑪𝑪

Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14 26

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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 above, we now are given that = . Since all radii of a circle are congruent, . Therefore, by SAS. because corresponding sides of congruent triangles are congruent 𝒎𝒎∠𝑨𝑨𝑨𝑨𝑨𝑨 𝒎𝒎 ∠𝑪𝑪𝑪𝑪𝑪𝑪 �𝑨𝑨��𝑨𝑨� ≅ 𝑩𝑩𝑩𝑩����� ≅ �𝑪𝑪��𝑪𝑪� ≅ �𝑫𝑫��𝑫𝑫�� △ 𝑨𝑨𝑨𝑨𝑨𝑨 ≅△ 𝑪𝑪𝑪𝑪𝑪𝑪 �𝑨𝑨𝑨𝑨��� ≅ �𝑪𝑪��𝑪𝑪�

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 is equidistant from the two chords. Two congruent chords equidistance If the center is equidistant from two from center chords, then the two chords are congruent.

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

Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14 27

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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 is said to be equidistant from two different points and if = .

𝑨𝑨 𝑩𝑩 𝑪𝑪 𝑨𝑨𝑨𝑨 𝑨𝑨𝑨𝑨

Exit Ticket (5 minutes)

Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14 28

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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 shown, = and = 22. Find .

𝐴𝐴 𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵 𝐷𝐷𝐷𝐷

2. In the figure, circle has a radius of 10. . a. If = 8, what is the length of ? 𝑃𝑃 �𝐴𝐴��𝐴𝐴� ⊥ �𝐷𝐷𝐷𝐷���

𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴

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

𝐷𝐷𝐷𝐷 𝐴𝐴𝐴𝐴

Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14 29

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY

Exit Ticket Sample Solutions

1. Given circle shown, = and = . Find .

𝑨𝑨 𝑨𝑨𝑨𝑨 𝑨𝑨𝑨𝑨 𝑩𝑩𝑩𝑩 𝟐𝟐𝟐𝟐 𝑫𝑫𝑫𝑫

𝟐𝟐𝟐𝟐

2. In the figure, circle has a radius of . . a. If = , what is the length of ? 𝑷𝑷 𝟏𝟏𝟏𝟏 �𝑨𝑨𝑨𝑨� �� ⊥ 𝑫𝑫𝑫𝑫���� 𝑨𝑨𝑨𝑨 𝟖𝟖 𝑨𝑨𝑨𝑨

𝟒𝟒 b. If = , what is the length of ?

𝑫𝑫 𝑫𝑫 𝟐𝟐 𝑨𝑨𝑨𝑨

𝟏𝟏 𝟏𝟏

Problem Set Sample Solutions

1. In this drawing, = , = , and = . What is ?

( . 𝑨𝑨𝑨𝑨) 𝟑𝟑𝟑𝟑 𝑶𝑶𝑶𝑶 𝟐𝟐𝟐𝟐 𝑶𝑶𝑶𝑶 𝟏𝟏𝟏𝟏 𝑪𝑪𝑪𝑪

𝟐𝟐 √𝟑𝟑𝟑𝟑𝟑𝟑 ≈ 𝟑𝟑𝟑𝟑 𝟕𝟕

2. In the figure to the right, and ; = . Find .

�𝑨𝑨��𝑨𝑨� ⊥ 𝑩𝑩𝑩𝑩���� �𝑫𝑫��𝑫𝑫� ⊥ 𝑬𝑬𝑬𝑬���� 𝑬𝑬𝑬𝑬 𝟏𝟏𝟏𝟏 𝑨𝑨𝑨𝑨

𝟐𝟐 𝟐𝟐

3. In the figure, = , and = . Find . Explain your work.

, is a𝑨𝑨 right𝑨𝑨 𝟐𝟐triangle𝟐𝟐 𝑫𝑫with𝑫𝑫 𝟏𝟏𝟏𝟏 𝑬𝑬𝑬𝑬= = and = , so = by . = = . 𝟓𝟓 △ 𝑨𝑨𝑨𝑨𝑨𝑨 𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡 𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫 𝟏𝟏𝟏𝟏 𝑨𝑨𝑨𝑨 𝟏𝟏𝟏𝟏 𝑩𝑩𝑩𝑩 𝟓𝟓 𝑩𝑩𝑩𝑩 𝑮𝑮𝑮𝑮 𝟓𝟓

Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14 30

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY

4. In the figure, = , = . Find .

𝑨𝑨𝑨𝑨 𝟏𝟏𝟏𝟏 𝑨𝑨𝑨𝑨 𝟏𝟏𝟏𝟏 𝑫𝑫𝑫𝑫

𝟒𝟒

5. In the figure, = , and the two concentric circles have radii of and . Find . 𝑪𝑪𝑪𝑪 𝟖𝟖 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝑫𝑫𝑫𝑫

𝟗𝟗

6. In the figure, the two circles have equal radii and intersect at points and . and are centers of the circles. = , and the radius of each circle is . . Find . Explain your work. 𝑩𝑩 𝑫𝑫 𝑨𝑨 𝑪𝑪 𝑨𝑨𝑨𝑨 𝟖𝟖 𝟓𝟓 �𝑩𝑩𝑩𝑩���� ⊥ 𝑨𝑨���𝑨𝑨� 𝑩𝑩𝑩𝑩

𝟔𝟔 = = (radii) 𝑩𝑩𝑩𝑩 = 𝑩𝑩𝑩𝑩 = 𝟓𝟓

𝑨𝑨𝑨𝑨 = 𝑮𝑮𝑮𝑮 (Pythagorean𝟒𝟒 theorem) 𝑩𝑩𝑩𝑩 = 𝟑𝟑

𝑩𝑩𝑩𝑩 𝟔𝟔

7. In the figure, the two concentric circles have radii of and . Chord of the larger circle intersects the smaller circle at and . = . . 𝟔𝟔 𝟏𝟏𝟏𝟏 �𝑩𝑩𝑩𝑩��� a. Find . 𝑪𝑪 𝑬𝑬 𝑪𝑪𝑪𝑪 𝟖𝟖 �𝑨𝑨��𝑨𝑨� ⊥ 𝑩𝑩𝑩𝑩���� 𝑨𝑨𝑨𝑨

𝟐𝟐 √𝟓𝟓 b. Find .

𝑩𝑩𝑩𝑩

𝟖𝟖 √𝟏𝟏𝟏𝟏

Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14 31

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY

8. In the figure, is the center of the circle, and = . Prove that bisects .

Let and 𝑨𝑨 be from to 𝑪𝑪 𝑪𝑪and 𝑪𝑪𝑪𝑪, respectively.�𝑨𝑨� �𝑨𝑨� ∠𝑩𝑩𝑩𝑩𝑩𝑩

�𝑨𝑨=��𝑨𝑨� �𝑨𝑨𝑨𝑨��� Given 𝑨𝑨 �𝑪𝑪��𝑪𝑪� �𝑪𝑪��𝑪𝑪�

𝑪𝑪𝑪𝑪 = 𝑪𝑪𝑪𝑪 If two chords are congruent, then the center is equidistant from the two chords. 𝑨𝑨𝑨𝑨 𝑨𝑨𝑨𝑨 = Reflexive property

𝑨𝑨𝑨𝑨 𝑨𝑨=𝑨𝑨 = ° A line joining the center and the midpoint of a chord is perpendicular to the chord.

∠𝑪𝑪𝑪𝑪𝑪𝑪 ∠𝑪𝑪𝑪𝑪𝑪𝑪 𝟗𝟗𝟗𝟗 HL

△ 𝑪𝑪𝑪𝑪𝑪𝑪=≅△ 𝑪𝑪𝑪𝑪𝑪𝑪 Corresponding angles of congruent triangles are equal in measure.

∠𝑬𝑬𝑬𝑬𝑬𝑬 bisects∠ 𝑭𝑭𝑭𝑭𝑭𝑭 Definition of angle bisector

𝑨𝑨 ���𝑨𝑨� ∠𝑩𝑩𝑩𝑩𝑩𝑩 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 ( and ) are congruent. Therefore, a rigid motion exists that carries to . The same rotation that carries to also carries to and to . The angle of ���� ���� rotation is the measure of , and𝑨𝑨𝑨𝑨 the rotation𝑪𝑪𝑪𝑪 is clockwise. �𝑨𝑨𝑨𝑨��� �𝑪𝑪��𝑪𝑪� 𝑨𝑨𝑨𝑨���� �𝑪𝑪��𝑫𝑫� 𝑨𝑨���𝑨𝑨� �𝑪𝑪��𝑪𝑪� �𝑩𝑩𝑩𝑩���� �𝑫𝑫��𝑫𝑫�� ∠𝑨𝑨𝑨𝑨𝑨𝑨 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 . Therefore, a rigid motion (a rotation) carries to . This same rotation carries to and to . The angle of rotation is the measure of , and the rotation is clockwise. ∠𝑨𝑨𝑨𝑨𝑨𝑨 ≅ ∠𝑪𝑪𝑪𝑪𝑪𝑪 ∠𝑨𝑨𝑨𝑨𝑨𝑨 ∠𝑪𝑪𝑪𝑪𝑪𝑪 �𝑨𝑨��𝑨𝑨� �𝑪𝑪��𝑪𝑪� �𝑩𝑩𝑩𝑩���� 𝑫𝑫���𝑫𝑫�� ∠𝑨𝑨𝑨𝑨𝑨𝑨

Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14 32

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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 Date: 10/22/14 33

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