Lesson 1: Thales' Theorem

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M5 GEOMETRY

Lesson 1: Thales’ Theorem

Classwork Opening Exercise a. Mark points and on the sheet of white paper provided by your teacher. b. Take the colored paper provided, and push that paper up between points and on the white sheet. 𝐴𝐴 𝐵𝐵 c. Mark on the white paper the location of the corner of the colored paper, using a different color than black. 𝐴𝐴 𝐵𝐵 Mark that point . See the example below.

𝐶𝐶 C

A B

d. Do this again, pushing the corner of the colored paper up between the black points but at a different angle. Again, mark the location of the corner. Mark this point . e. Do this again and then again, multiple times. Continue to label the points. What curve do the colored points 𝐷𝐷 ( , , …) seem to trace?

𝐶𝐶 𝐷𝐷 Exploratory Challenge Choose one of the colored points ( , , ...) that you marked. Draw the right triangle formed by the line segment connecting the original two points and and that colored point. Take a copy of the triangle, and rotate it 180° about the midpoint of . 𝐶𝐶 𝐷𝐷 𝐴𝐴 𝐵𝐵 Label the acute angles�𝐴𝐴��𝐴𝐴� in the original triangle as and , and label the corresponding angles in the rotated triangle the same. 𝑥𝑥 𝑦𝑦 Todd says is a rectangle. Maryam says is a quadrilateral, but she is not sure it is a rectangle. Todd is right but does not know how to explain himself to Maryam. Can you help him out? 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴′ 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴′ a. What composite figure is formed by the two triangles? How would you prove it?

i. What is the sum of the measures of and ? Why?

𝑥𝑥 𝑦𝑦

Lesson 1: Thales’ Theorem S.1

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M5 GEOMETRY

ii. How do we know that the figure whose vertices are the colored points ( , , …) and points and is a rectangle? 𝐶𝐶 𝐷𝐷 𝐴𝐴 𝐵𝐵

b. Draw the two diagonals of the rectangle. Where is the midpoint of the segment connecting the two original points and ? Why?

𝐴𝐴 𝐵𝐵

c. Label the intersection of the diagonals as point . How does the distance from point to a colored point ( , , …) compare to the distance from to points and ? 𝑃𝑃 𝑃𝑃 𝐶𝐶 𝐷𝐷 𝑃𝑃 𝐴𝐴 𝐵𝐵

d. Choose another colored point, and construct a rectangle using the same process you followed before. Draw the two diagonals of the new rectangle. How do the diagonals of the new and old rectangle compare? How do you know?

e. How does your drawing demonstrate that all the colored points you marked do indeed lie on a circle?

Lesson 1: Thales’ Theorem S.2

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M5 GEOMETRY

Example

In the Exploratory Challenge, you proved the converse of a famous theorem in geometry. Thales’ theorem states the following: If , , and are three distinct points on a circle, and is a diameter of the circle, then is right.

Notice that, in𝐴𝐴 the𝐵𝐵 proof𝐶𝐶 in the Exploratory Challenge, you started�𝐴𝐴� with�𝐴𝐴� a right angle (the corner of the∠𝐴𝐴 colored𝐴𝐴𝐴𝐴 paper) and created a circle. With Thales’ theorem, you must start with the circle and then create a right angle. Prove Thales’ theorem. a. Draw circle with distinct points , , and on the circle and diameter . Prove that is a right angle.

𝑃𝑃 𝐴𝐴 𝐵𝐵 𝐶𝐶 �𝐴𝐴��𝐴𝐴� ∠𝐴𝐴𝐴𝐴𝐴𝐴

b. Draw a third radius ( ). What types of triangles are and ? How do you know?

�𝑃𝑃��𝑃𝑃� △ 𝐴𝐴𝐴𝐴𝐴𝐴 △ 𝐵𝐵𝐵𝐵𝐵𝐵

c. Using the diagram that you just created, develop a strategy to prove Thales’ theorem.

d. Label the base angles of as ° and the base angles of as °. Express the measure of in terms of ° and °. △ 𝐴𝐴𝐴𝐴𝐴𝐴 𝑏𝑏 △ 𝐵𝐵𝐵𝐵𝐵𝐵 𝑎𝑎 ∠𝐴𝐴𝐴𝐴𝐴𝐴 𝑎𝑎 𝑏𝑏

e. How can the previous conclusion be used to prove that is a right angle?

∠𝐴𝐴𝐴𝐴𝐴𝐴

Lesson 1: Thales’ Theorem S.3

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M5 GEOMETRY

Exercises 1. is a diameter of the circle shown. The radius is 12.5 cm, and = 7 cm. a. Find . �𝐴𝐴��𝐴𝐴� 𝐴𝐴𝐴𝐴 𝑚𝑚∠𝐶𝐶

b. Find .

𝐴𝐴𝐴𝐴

c. Find .

𝐵𝐵𝐵𝐵

2. In the circle shown, is a diameter with center . a. Find . �𝐵𝐵��𝐵𝐵� 𝐴𝐴 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷

b. Find .

𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵

c. Find .

𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷

Lesson 1: Thales’ Theorem S.4

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M5 GEOMETRY

Lesson Summary

Theorems:

. THALES’ THEOREM: If , , and are three different points on a circle with a diameter , then is a right angle. 𝐴𝐴 𝐵𝐵 𝐶𝐶 �𝐴𝐴��𝐴𝐴� ∠𝐴𝐴𝐴𝐴𝐴𝐴 . CONVERSE OF THALES’ THEOREM: If is a right triangle with the right angle, then , , and are three distinct points on a circle with a diameter . △ 𝐴𝐴𝐴𝐴𝐴𝐴 ∠𝐶𝐶 𝐴𝐴 𝐵𝐵 𝐶𝐶 . Therefore, given distinct points , , and on a circle, is a right triangle with the right angle if �𝐴𝐴��𝐴𝐴� and only if is a diameter of the circle. 𝐴𝐴 𝐵𝐵 𝐶𝐶 △ 𝐴𝐴𝐴𝐴𝐴𝐴 ∠𝐶𝐶 . Given two points and , let point be the midpoint between them. If is a point such that is �𝐴𝐴��𝐴𝐴� right, then = = . 𝐴𝐴 𝐵𝐵 𝑃𝑃 𝐶𝐶 ∠𝐴𝐴𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵 𝐴𝐴𝐴𝐴 𝐶𝐶𝐶𝐶 Relevant Vocabulary

. CIRCLE: Given a point in the plane and a number > 0, the circle with center and radius is the set of all points in the plane that are distance from the point . 𝐶𝐶 𝑟𝑟 𝐶𝐶 𝑟𝑟 . RADIUS: May refer either to the line segment joining the center of a circle with any point on that circle (a 𝑟𝑟 𝐶𝐶 radius) or to the length of this line segment (the radius). . DIAMETER: May refer either to the segment that passes through the center of a circle whose endpoints lie on the circle (a diameter) or to the length of this line segment (the diameter). . CHORD: Given a circle , and let and be points on . is called a chord of . . CENTRAL ANGLE: A central angle of a circle is an angle whose vertex is the center of a circle. 𝐶𝐶 𝑃𝑃 𝑄𝑄 𝐶𝐶 �𝑃𝑃��𝑃𝑃� 𝐶𝐶

Problem Set

1. , , and are three points on a circle, and angle is a right angle. What is wrong with the picture below? Explain your reasoning. 𝐴𝐴 𝐵𝐵 𝐶𝐶 𝐴𝐴𝐴𝐴𝐴𝐴

Lesson 1: Thales’ Theorem S.5

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M5 GEOMETRY

2. Show that there is something mathematically wrong with the picture below.

3. In the figure below, is the diameter of a circle of radius 17 miles. If = 30 miles, what is ?

�𝐴𝐴��𝐴𝐴� 𝐵𝐵𝐵𝐵 𝐴𝐴𝐴𝐴

4. In the figure below, is the center of the circle, and is a diameter.

𝑂𝑂 �𝐴𝐴��𝐴𝐴�

a. Find . b. If = 3 4, what is ? 𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴 𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴 ∶ 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 ∶ 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵

Lesson 1: Thales’ Theorem S.6

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M5 GEOMETRY

5. is a diameter of a circle, and is another point on the circle. The point lies on such that = . Show that = . (Hint: Draw a picture to help you explain your thinking.) �𝑃𝑃��𝑃𝑃� 𝑀𝑀 𝑅𝑅 ⃖𝑀𝑀��𝑀𝑀�⃗

𝑅𝑅𝑅𝑅 𝑀𝑀𝑀𝑀 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃 6. Inscribe in a circle of diameter 1 such that is a diameter. Explain why: a. sin( ) = . △ 𝐴𝐴𝐴𝐴𝐴𝐴 �𝐴𝐴��𝐴𝐴� b. cos( ) = . ∠𝐴𝐴 𝐵𝐵𝐵𝐵 ∠𝐴𝐴 𝐴𝐴𝐴𝐴

Lesson 1: Thales’ Theorem S.7

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 below (as you did in Module 1).

�𝐴𝐴𝐴𝐴��

Draw another line that bisects but is not perpendicular to it.

List one similarity and one difference�𝐴𝐴𝐴𝐴�� between the two bisectors.

Lesson 2: Circles, Chords, Diameters, and Their Relationships S.8

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 S.9

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 S.10

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 S.11

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 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 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.

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 S.12

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

Problem Set

Figures are not drawn to scale.

1. In this drawing, = 30, = 20, and = 18. What is ?

𝐴𝐴𝐴𝐴 𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂 𝐶𝐶𝐶𝐶

2. In the figure to the right, , , and = 12. Find .

𝐴𝐴��𝐴𝐴� ⊥ 𝐵𝐵��𝐵𝐵� 𝐷𝐷𝐷𝐷�� ⊥ 𝐸𝐸��𝐸𝐸� 𝐸𝐸𝐸𝐸 𝐴𝐴𝐴𝐴

Lesson 2: Circles, Chords, Diameters, and Their Relationships S.13

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY

3. In the figure, = 24, and = 13. Find . Explain your work.

𝐴𝐴𝐴𝐴 𝐷𝐷𝐷𝐷 𝐸𝐸𝐸𝐸

4. In the figure, = 10, and = 16. Find .

𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴 𝐷𝐷𝐷𝐷

5. In the figure, = 8, and the two concentric circles have radii of 10 and 17. Find .

𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷

6. In the figure, the two circles have equal radii and intersect at points and . and are centers of the circles. = 8, and the radius of each circle is 5. . Find . Explain your work. 𝐵𝐵 𝐷𝐷 𝐴𝐴 𝐶𝐶 𝐴𝐴 𝐴𝐴 �𝐵𝐵𝐵𝐵�� ⊥ �𝐴𝐴��𝐴𝐴� 𝐵𝐵𝐵𝐵

Lesson 2: Circles, Chords, Diameters, and Their Relationships S.14

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY

7. In the figure, the two concentric circles have radii of 6 and 14. Chord of the larger circle intersects the smaller circle at and . = 8. . �𝐵𝐵𝐵𝐵�� a. Find . 𝐶𝐶 𝐸𝐸 𝐶𝐶𝐶𝐶 �𝐴𝐴��𝐴𝐴� ⊥ �𝐵𝐵𝐵𝐵�� b. Find . 𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵

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

𝐴𝐴 𝐶𝐶𝐶𝐶 𝐶𝐶𝐶𝐶 �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. 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 S.15

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY

Lesson 3: Rectangles Inscribed in Circles

Classwork Opening Exercise Using only a compass and straightedge, find the location of the center of the circle below. Follow the steps provided. . Draw chord . . Construct a chord perpendicular to at endpoint . �𝐴𝐴𝐴𝐴�� . Mark the point of intersection of the perpendicular chord and the circle �𝐴𝐴𝐴𝐴�� 𝐵𝐵 as point . . is a diameter of the circle. Construct a second diameter in the same 𝐶𝐶 way. 𝐴𝐴��𝐴𝐴� . Where the two diameters meet is the center of the circle.

Explain why the steps of this construction work.

Exploratory Challenge Construct a rectangle such that all four vertices of the rectangle lie on the circle below.

Lesson 3: Rectangles Inscribed in Circles S.16

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY

Exercises 1. Construct a kite inscribed in the circle below, and explain the construction using symmetry.

2. Given a circle and a rectangle, what must be true about the rectangle for it to be possible to inscribe a congruent copy of it in the circle?

3. The figure below shows a rectangle inscribed in a circle.

a. List the properties of a rectangle.

Lesson 3: Rectangles Inscribed in Circles S.17

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY

b. List all the symmetries this diagram possesses.

c. List the properties of a square.

d. List all the symmetries of the diagram of a square inscribed in a circle.

4. A rectangle is inscribed into a circle. The rectangle is cut along one of its diagonals and reflected across that diagonal to form a kite. Draw the kite and its diagonals. Find all the angles in this new diagram, given that the acute angle formed by the diagonal of the rectangle in the original diagram was 40°.

Lesson 3: Rectangles Inscribed in Circles S.18

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY

5. Challenge: Show that the three vertices of a right triangle are equidistant from the midpoint of the hypotenuse by showing that the perpendicular bisectors of the legs pass through the midpoint of the hypotenuse. a. Draw the perpendicular bisectors of and .

�𝐴𝐴𝐴𝐴�� 𝐴𝐴��𝐴𝐴� b. Label the point where they meet . What is point ?

𝑃𝑃 𝑃𝑃

c. What can be said about the distance from to each vertex of the triangle? What is the relationship between the circle and the triangle? 𝑃𝑃

d. Repeat this process, this time sliding to another place on the circle and call it . What do you notice?

𝐵𝐵 𝐵𝐵′

e. Is there a relationship between and ? Explain.

𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴 𝑚𝑚∠𝐴𝐴𝐴𝐴′𝐶𝐶

Lesson 3: Rectangles Inscribed in Circles S.19

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY

Lesson Summary

Relevant Vocabulary

INSCRIBED POLYGON: A polygon is inscribed in a circle if all vertices of the polygon lie on the circle.

Problem Set

Figures are not drawn to scale. 1. Using only a piece of 8.5 × 11 inch copy paper and a pencil, find the location of the center of the circle below.

2. Is it possible to inscribe a parallelogram that is not a rectangle in a circle?

3. In the figure, is a rectangle inscribed in circle . = 8; = 12. Find .

𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝐴𝐴 𝐷𝐷𝐷𝐷 𝐵𝐵𝐵𝐵 𝐴𝐴𝐴𝐴

4. Given the figure, = = 8 and = 13. Find the radius of the circle.

𝐵𝐵𝐵𝐵 𝐶𝐶𝐶𝐶 𝐴𝐴𝐴𝐴

Lesson 3: Rectangles Inscribed in Circles S.20

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY

5. In the figure, and are parallel chords 14 cm apart. = 12 cm, = 10 cm, and . Find .

𝐷𝐷��𝐷𝐷� 𝐵𝐵��𝐵𝐵� 𝐷𝐷𝐷𝐷 𝐴𝐴𝐴𝐴 �𝐸𝐸��𝐸𝐸� ⊥ �𝐵𝐵��𝐵𝐵� 𝐵𝐵𝐵𝐵

6. Use perpendicular bisectors of the sides of a triangle to construct a circle that circumscribes the triangle.

Lesson 3: Rectangles Inscribed in Circles S.21

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 M5 GEOMETRY

Lesson 4: Experiments with Inscribed Angles

Classwork Opening Exercise

ARC:

MINOR AND MAJOR ARC:

INSCRIBED ANGLE:

CENTRAL ANGLE:

INTERCEPTED ARC OF AN ANGLE:

Lesson 4: Experiments with Inscribed Angles S.22

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 M5 GEOMETRY

Exploratory Challenge 1 Your teacher will provide you with a straightedge, a sheet of colored paper in the shape of a trapezoid, and a sheet of plain white paper. . Draw two points no more than 3 inches apart in the middle of the plain white paper, and label them and . . Use the acute angle of your colored trapezoid to plot a point on the white sheet by placing the colored cutout 𝐴𝐴 𝐵𝐵 so that the points and are on the edges of the acute angle and then plotting the position of the vertex of the angle. Label that vertex . 𝐴𝐴 𝐵𝐵 . Repeat several times. Name the points , , …. 𝐶𝐶 𝐷𝐷 𝐸𝐸 Exploratory Challenge 2 a. Draw several of the angles formed by connecting points and on your paper with any of the additional points you marked as the acute angle was pushed through the points ( , , , …). What do you notice about the measures of these angles? 𝐴𝐴 𝐵𝐵 𝐶𝐶 𝐷𝐷 𝐸𝐸

b. Draw several of the angles formed by connecting points and on your paper with any of the additional points you marked as the obtuse angle was pushed through the points from above. What do you notice about the measures of these angles? 𝐴𝐴 𝐵𝐵

Exploratory Challenge 3 a. Draw a point on the circle, and label it . Create angle .

𝐷𝐷 ∠𝐵𝐵𝐵𝐵𝐵𝐵 b. is called an inscribed angle. Can you explain why?

∠𝐵𝐵𝐵𝐵𝐵𝐵

Lesson 4: Experiments with Inscribed Angles S.23

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 M5 GEOMETRY

c. Arc is called the intercepted arc. Can you explain why?

𝐵𝐵𝐵𝐵

d. Carefully cut out the inscribed angle, and compare it to the angles of several of your neighbors.

e. What appears to be true about each of the angles you drew?

f. Draw another point on a second circle, and label it point . Create , and cut it out. Compare and . What appears to be true about the two angles? 𝐸𝐸 ∠𝐵𝐵𝐵𝐵𝐵𝐵 ∠𝐵𝐵𝐵𝐵𝐵𝐵 ∠𝐵𝐵𝐵𝐵𝐵𝐵

g. What conclusion may be drawn from this? Will all angles inscribed in the circle from these two points have the same measure?

h. Explain to your neighbor what you have just discovered.

Lesson 4: Experiments with Inscribed Angles S.24

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 M5 GEOMETRY

Exploratory Challenge 4 a. In the circle below, draw the angle formed by connecting points and to the center of the circle.

𝐵𝐵 𝐶𝐶

b. Is an inscribed angle? Explain.

∠𝐵𝐵𝐵𝐵𝐵𝐵

c. Is it appropriate to call this the central angle? Why or why not?

d. What is the intercepted arc?

e. Is the measure of the same as the measure of one of the inscribed angles in Exploratory Challenge 2?

∠𝐵𝐵𝐵𝐵𝐵𝐵

f. Can you make a prediction about the relationship between the inscribed angle and the central angle?

Lesson 4: Experiments with Inscribed Angles S.25

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 M5 GEOMETRY

Lesson Summary

All inscribed angles from the same intercepted arc have the same measure. Relevant Vocabulary

. ARC: An arc is a portion of the circumference of a circle. . MINOR AND MAJOR ARC: Let be a circle with center , and let and be different points that lie on but are not the endpoints of the same diameter. The minor arc is the set containing , , and all points of that are in the interior of 𝐶𝐶 . The major arc is the𝑂𝑂 set containing𝐴𝐴 𝐵𝐵 , , and all points of that lie𝐶𝐶 in the exterior of . 𝐴𝐴 𝐵𝐵 𝐶𝐶 ∠𝐴𝐴𝐴𝐴𝐴𝐴 𝐴𝐴 𝐵𝐵 𝐶𝐶 . INSCRIBED ANGLE: An inscribed angle is an angle whose vertex is on a circle, and each side of the angle ∠𝐴𝐴𝐴𝐴𝐴𝐴 intersects the circle in another point. . CENTRAL ANGLE: A central angle of a circle is an angle whose vertex is the center of a circle. . INTERCEPTED ARC OF AN ANGLE: An angle intercepts an arc if the endpoints of the arc lie on the angle, all other points of the arc are in the interior of the angle, and each side of the angle contains an endpoint of the arc.

Problem Set

1. Using a protractor, measure both the inscribed angle and the central angle shown on the circle below.

= ______= ______

𝑚𝑚 ∠𝐵𝐵𝐵𝐵𝐵𝐵 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 M5 GEOMETRY

2. Using a protractor, measure both the inscribed angle and the central angle shown on the circle below.

= ______= ______

𝑚𝑚 ∠𝐵𝐵𝐵𝐵𝐵𝐵 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 3. Using a protractor, measure both the inscribed angle and the central angle shown on the circle below.

= ______= ______

𝑚𝑚 ∠𝐵𝐵𝐵𝐵𝐵𝐵 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 4. What relationship between the measure of the inscribed angle and the measure of the central angle that intercepts the same arc is illustrated by these examples?

5. Is your conjecture at least true for inscribed angles that measure 90°?

6. Prove that = 2 in the diagram below.

𝑦𝑦 𝑥𝑥

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 M5 GEOMETRY

7. Red ( ) and blue ( ) lighthouses are located on the coast of the ocean. Ships traveling are in safe waters as long as the angle from the ship ( ) to the two lighthouses ( ) is always less than or equal to some angle called the danger𝑅𝑅 angle. What𝐵𝐵 happens to as the ship gets closer to shore and moves away from shore? Why do you think a larger angle is dangerous?𝑆𝑆 ∠𝑅𝑅𝑅𝑅𝑅𝑅 𝜃𝜃 𝜃𝜃

Red ( )

𝑹𝑹

Blue ( )

𝑩𝑩

Lesson 4: Experiments with Inscribed Angles S.28

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M5 GEOMETRY

Lesson 5: Inscribed Angle Theorem and Its Applications

Classwork Opening Exercise a. and are points on a circle with center . i. Draw a point on the circle so that is a diameter. 𝐴𝐴 𝐶𝐶 𝑂𝑂 Then draw the angle . 𝐵𝐵 �𝐴𝐴��𝐴𝐴�

𝐴𝐴𝐴𝐴𝐴𝐴 ii. What angle in your diagram is an inscribed angle?

iii. What angle in your diagram is a central angle?

iv. What is the intercepted arc of ?

∠𝐴𝐴𝐴𝐴𝐴𝐴

v. What is the intercepted arc of ?

∠𝐴𝐴𝐴𝐴𝐴𝐴

b. The measure of the inscribed angle is , and the measure of the central angle is . Find in terms of . 𝐴𝐴𝐴𝐴𝐴𝐴 𝑥𝑥 𝐶𝐶𝐶𝐶𝐶𝐶 𝑦𝑦 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 𝑥𝑥

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M5 GEOMETRY

Example 1

and are points on a circle with center .

𝐴𝐴 𝐶𝐶 𝑂𝑂

a. What is the intercepted arc of ? Color it red.

∠𝐶𝐶𝐶𝐶𝐶𝐶

b. Draw triangle . What type of triangle is it? Why?

𝐴𝐴𝐴𝐴𝐴𝐴

c. What can you conclude about and ? Why?

𝑚𝑚∠𝑂𝑂𝑂𝑂𝑂𝑂 𝑚𝑚∠𝑂𝑂𝑂𝑂𝑂𝑂

d. Draw a point on the circle so that is in the interior of the inscribed angle .

𝐵𝐵 𝑂𝑂 𝐴𝐴𝐴𝐴𝐴𝐴 e. What is the intercepted arc of ? Color it green.

∠𝐴𝐴𝐴𝐴𝐴𝐴

f. What do you notice about ?

𝐴𝐴�𝐴𝐴

Lesson 5: Inscribed Angle Theorem and Its Applications S.30

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M5 GEOMETRY

g. Let the measure of be and the measure of be . Can you prove that = 2 ? (Hint: Draw the diameter that contains point .) ∠𝐴𝐴𝐴𝐴𝐴𝐴 𝑥𝑥 ∠𝐴𝐴𝐴𝐴𝐴𝐴 𝑦𝑦 𝑦𝑦 𝑥𝑥

𝐵𝐵

h. Does your conclusion support the inscribed angle theorem?

i. If we combine the Opening Exercise and this proof, have we finished proving the inscribed angle theorem?

Example 2

and are points on a circle with center .

𝐴𝐴 𝐶𝐶 𝑂𝑂

a. Draw a point on the circle so that is in the exterior of the inscribed angle .

𝐵𝐵 𝑂𝑂 𝐴𝐴𝐴𝐴𝐴𝐴 b. What is the intercepted arc of ? Color it yellow.

∠𝐴𝐴𝐴𝐴𝐴𝐴

c. Let the measure of be and the measure of be . Can you prove that = 2 ? (Hint: Draw the diameter that contains point .) ∠𝐴𝐴𝐴𝐴𝐴𝐴 𝑥𝑥 ∠𝐴𝐴𝐴𝐴𝐴𝐴 𝑦𝑦 𝑦𝑦 𝑥𝑥

𝐵𝐵

Lesson 5: Inscribed Angle Theorem and Its Applications S.31

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M5 GEOMETRY

d. Does your conclusion support the inscribed angle theorem?

e. Have we finished proving the inscribed angle theorem?

Exercises 1. Find the measure of the angle with measure . Diagrams are not drawn to scale.

a. = 25° b. 𝑥𝑥 = 15° c. = 90° 𝑚𝑚∠𝐷𝐷 𝑚𝑚∠𝐷𝐷 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 x

x x

Lesson 5: Inscribed Angle Theorem and Its Applications S.32

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M5 GEOMETRY

d. = 32° e. = f. = 19°

𝑚𝑚∠𝐵𝐵 𝐵𝐵𝐵𝐵 𝐴𝐴𝐴𝐴 𝑚𝑚∠𝐷𝐷

2. Toby says is a right triangle because = 90°. Is he correct? Justify your answer.

△ 𝐵𝐵𝐵𝐵𝐵𝐵 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵

Lesson 5: Inscribed Angle Theorem and Its Applications S.33

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M5 GEOMETRY

3. Let’s look at relationships between inscribed angles. a. Examine the inscribed polygon below. Express in terms of and in terms of . Are the opposite angles in any quadrilateral inscribed in a circle supplementary? Explain. 𝑥𝑥 𝑦𝑦 𝑦𝑦 𝑥𝑥

b. Examine the diagram below. How many angles have the same measure, and what are their measures in terms of ?

𝑥𝑥˚

Lesson 5: Inscribed Angle Theorem and Its Applications S.34

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M5 GEOMETRY

4. Find the measures of the labeled angles. a. b.

c. d.

e. f.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M5 GEOMETRY

Lesson Summary

Theorems:

Relevant Vocabulary

. INSCRIBED ANGLE: An inscribed angle is an angle whose vertex is on a circle, and each side of the angle intersects the circle in another point. . INTERCEPTED ARC: An angle intercepts an arc if the endpoints of the arc lie on the angle, all other points of the arc are in the interior of the angle, and each side of the angle contains an endpoint of the arc. An angle inscribed in a circle intercepts exactly one arc, in particular, the arc intercepted by an inscribed right angle is the semicircle in the interior of the angle.

Problem Set

For Problems 1–8, find the value of . Diagrams are not drawn to scale.

1. 𝑥𝑥 2.

Lesson 5: Inscribed Angle Theorem and Its Applications S.36

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M5 GEOMETRY

3. 4.

5. 6.

7. 8.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M5 GEOMETRY

9. a. The two circles shown intersect at and . The center of the larger circle, , lies on the circumference of the smaller circle. If a chord of the larger circle, , cuts the smaller circle at , find and . 𝐸𝐸 𝐹𝐹 𝐷𝐷 �𝐹𝐹𝐹𝐹�� 𝐻𝐻 𝑥𝑥 𝑦𝑦

b. How does this problem confirm the inscribed angle theorem?

10. In the figure below, and intersect at point .

Prove: + �𝐸𝐸𝐸𝐸�� =�𝐵𝐵��𝐵𝐵�2( ) 𝐹𝐹 PROOF: 𝑚𝑚 ∠𝐷𝐷𝐷𝐷𝐷𝐷Join 𝑚𝑚.∠ 𝐸𝐸𝐸𝐸𝐸𝐸 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵�� 1 = ( ______) 2 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 𝑚𝑚∠ 1 = ( ______) 2 𝑚𝑚∠𝐸𝐸𝐸𝐸𝐸𝐸 𝑚𝑚∠ In ,

∆𝐸𝐸𝐸𝐸𝐸𝐸 + = ______

𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 𝑚𝑚∠𝐸𝐸𝐸𝐸𝐸𝐸 𝑚𝑚∠ 1 1 ( ______) + ( ______) = ______2 2 𝑚𝑚∠ 𝑚𝑚∠ 𝑚𝑚∠ + = 2( )

∴ 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 𝑚𝑚∠𝐸𝐸𝐸𝐸𝐸𝐸 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵

Lesson 5: Inscribed Angle Theorem and Its Applications S.38

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M5 GEOMETRY

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles

Classwork Opening Exercise In a circle, a chord and a diameter are extended outside of the circle to meet at point . If = 46°, and = 32°, find . �𝐷𝐷𝐷𝐷�� 𝐴𝐴𝐴𝐴�� 𝐶𝐶 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷

Let = , =

In 𝑚𝑚∠𝐷𝐷𝐷𝐷, 𝐷𝐷 𝑦𝑦˚ 𝑚𝑚=∠ 𝐸𝐸𝐸𝐸𝐸𝐸 𝑥𝑥˚ Reason:

△ 𝐴𝐴𝐴𝐴𝐴𝐴=𝑚𝑚 ∠𝐷𝐷𝐷𝐷𝐷𝐷 Reason:

𝑚𝑚∠46𝐴𝐴𝐴𝐴+𝐴𝐴 + + 90 = Reason:

∴ + =𝑥𝑥 𝑦𝑦

In𝑥𝑥 𝑦𝑦 , = + 32 Reason:

+△ 𝐴𝐴+𝐴𝐴𝐴𝐴32𝑦𝑦= 𝑥𝑥 Reason:

𝑥𝑥 = 𝑥𝑥

𝑥𝑥 = 𝑦𝑦 = 𝑚𝑚 ∠𝐷𝐷𝐷𝐷𝐷𝐷

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles S.39

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M5 GEOMETRY

Exercises Find the value in each figure below, and describe how you arrived at the answer.

1. Hint: Thales’𝑥𝑥 theorem 2.

3. 4.

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles S.40

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M5 GEOMETRY

Lesson Summary

Theorems:

. THE INSCRIBED ANGLE THEOREM: The measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle. . CONSEQUENCE OF INSCRIBED ANGLE THEOREM: Inscribed angles that intercept the same arc are equal in measure. . If , , , and are four points with and ’ on the same side of , and and are congruent, then , , , and all lie on the same circle. 𝐴𝐴 𝐵𝐵 𝐵𝐵′ 𝐶𝐶 𝐵𝐵 𝐵𝐵 ⃖𝐴𝐴��𝐴𝐴�⃗ ∠𝐴𝐴𝐴𝐴𝐴𝐴 ∠𝐴𝐴𝐴𝐴′𝐶𝐶 𝐴𝐴 𝐵𝐵 𝐵𝐵′ 𝐶𝐶 Relevant Vocabulary

. CENTRAL ANGLE: A central angle of a circle is an angle whose vertex is the center of a circle. . INSCRIBED ANGLE: An inscribed angle is an angle whose vertex is on a circle, and each side of the angle intersects the circle in another point. . INTERCEPTED ARC: An angle intercepts an arc if the endpoints of the arc lie on the angle, all other points of the arc are in the interior of the angle, and each side of the angle contains an endpoint of the arc. An angle inscribed in a circle intercepts exactly one arc, in particular, the arc intercepted by a right angle is the semicircle in the interior of the angle.

Problem Set

In Problems 1–5, find the value .

1. 𝑥𝑥

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles S.41

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M5 GEOMETRY

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles S.42

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M5 GEOMETRY

6. If = , express in terms of .

𝐵𝐵𝐵𝐵 𝐹𝐹𝐹𝐹 𝑦𝑦 𝑥𝑥

7. a. Find the value of . b. Suppose the = °. Prove that = 3 °. 𝑥𝑥

𝑚𝑚∠𝐶𝐶 𝑎𝑎 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 𝑎𝑎

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles S.43

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M5 GEOMETRY

8. In the figure below, three identical circles meet at , , , and , respectively. = . , , and , , lie on straight lines. 𝐵𝐵 𝐹𝐹 𝐶𝐶 𝐸𝐸 𝐵𝐵𝐵𝐵 𝐶𝐶𝐶𝐶 𝐴𝐴 𝐵𝐵 𝐶𝐶 𝐹𝐹 𝐸𝐸 𝐷𝐷 Prove is a parallelogram.

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴

PROOF:

Join and .

=𝐵𝐵𝐵𝐵 𝐶𝐶𝐶𝐶 Reason: ______

𝐵𝐵 𝐵𝐵= ______𝐶𝐶𝐶𝐶 = ______= ______= Reason: ______𝑎𝑎______= ______𝑑𝑑 Alternate interior angles are equal in measure.

𝐴𝐴�______��𝐴𝐴� ∥ 𝐹𝐹𝐹𝐹�� = ______Corresponding angles are equal in measure.

𝐴𝐴�______��𝐴𝐴� ∥ 𝐵𝐵𝐵𝐵�� = ______Corresponding angles are equal in measure.

𝐵𝐵𝐵𝐵�� ∥ �𝐶𝐶��𝐶𝐶� 𝐴𝐴��𝐴𝐴� ∥ 𝐵𝐵𝐵𝐵� �is�� a∥ parallelogram.�𝐶𝐶��𝐶𝐶� 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles S.44

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M5 GEOMETRY

Lesson 7: The Angle Measure of an Arc

Classwork Opening Exercise If the measure of is 17°, name three other angles that have the same measure and explain why. ∠𝐺𝐺𝐺𝐺𝐺𝐺

What is the measure of ? Explain.

∠𝐺𝐺𝐺𝐺𝐺𝐺

Can you find the measure of ? Explain.

∠𝐵𝐵𝐵𝐵𝐵𝐵

Lesson 7: The Angle Measure of an Arc S.45

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M5 GEOMETRY

Example

What if we started with an angle inscribed in the minor arc between and ?

𝐴𝐴 𝐶𝐶

Exercises 1. In circle , : : : = 1: 2: 3: 4. Find the following angles of measure. a. 𝐴𝐴 𝑚𝑚𝐵𝐵𝐵𝐵� 𝑚𝑚𝐶𝐶𝐶𝐶� 𝑚𝑚𝐸𝐸𝐸𝐸� 𝑚𝑚𝐷𝐷𝐷𝐷�

𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵

𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷

𝑚𝑚𝐷𝐷𝐷𝐷�

𝑚𝑚𝐶𝐶𝐶𝐶�𝐶𝐶

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M5 GEOMETRY

2. In circle , = . Find the following angles of measure. a. 𝐵𝐵 𝐴𝐴𝐴𝐴 𝐶𝐶𝐶𝐶

𝑚𝑚𝐶𝐶�𝐶𝐶

𝑚𝑚𝐶𝐶�𝐶𝐶𝐶𝐶

𝑚𝑚𝐴𝐴𝐴𝐴𝐴𝐴�

3. In circle , is a diameter and = 100°. If = 2 , find the following angles of measure. a. 𝐴𝐴 �𝐵𝐵𝐵𝐵�� 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 𝑚𝑚𝐸𝐸𝐸𝐸� 𝑚𝑚𝐵𝐵�𝐵𝐵

𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵

𝑚𝑚𝐸𝐸𝐸𝐸�

𝑚𝑚𝐷𝐷𝐷𝐷𝐷𝐷�

4. Given circle with = 37°, find the following angles of measure. a. 𝐴𝐴 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶

𝑚𝑚𝐶𝐶�𝐶𝐶𝐶𝐶

𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M5 GEOMETRY

Lesson Summary

Theorems:

. INSCRIBED ANGLE THEOREM: The measure of an inscribed angle is half the measure of its intercepted arc. . Two arcs (of possibly different circles) are similar if they have the same angle measure. Two arcs in the same or congruent circles are congruent if they have the same angle measure. . All circles are similar.

Relevant Vocabulary

. ARC: An arc is a portion of the circumference of a circle. . MINOR AND MAJOR ARC: Let be a circle with center , and let and be different points that lie on but are not the endpoints of the same diameter. The minor arc is the set containing , , and all points of that are in the interior of 𝐶𝐶 . The major arc is the𝑂𝑂 set containing𝐴𝐴 𝐵𝐵 , , and all points of that lie𝐶𝐶 in the exterior of . 𝐴𝐴 𝐵𝐵 𝐶𝐶 ∠𝐴𝐴𝐴𝐴𝐴𝐴 𝐴𝐴 𝐵𝐵 𝐶𝐶 . SEMICIRCLE: In a circle, let and be the endpoints of a diameter. A semicircle is the set containing , , ∠𝐴𝐴𝐴𝐴𝐴𝐴 and all points of the circle that lie in a given half-plane of the line determined by the diameter. 𝐴𝐴 𝐵𝐵 𝐴𝐴 𝐵𝐵 . INSCRIBED ANGLE: An inscribed angle is an angle whose vertex is on a circle and each side of the angle intersects the circle in another point. . CENTRAL ANGLE: A central angle of a circle is an angle whose vertex is the center of a circle. . INTERCEPTED ARC OF AN ANGLE: An angle intercepts an arc if the endpoints of the arc lie on the angle, all other points of the arc are in the interior of the angle, and each side of the angle contains an endpoint of the arc.

Problem Set

1. Given circle with = 50°, a. Name a central angle. 𝐴𝐴 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 b. Name an inscribed angle. c. Name a chord. d. Name a minor arc. e. Name a major arc. f. Find . g. Find . 𝑚𝑚𝐶𝐶�𝐶𝐶 h. Find . 𝑚𝑚𝐶𝐶�𝐶𝐶𝐶𝐶

𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶

Lesson 7: The Angle Measure of an Arc S.48

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M5 GEOMETRY

2. Given circle , find the measure of each minor arc.

𝐴𝐴

3. Given circle , find the following measure. a. 𝐴𝐴 b. 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 c. 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 d. 𝑚𝑚𝐵𝐵𝐵𝐵� e. 𝑚𝑚𝐵𝐵�𝐵𝐵 𝑚𝑚𝐵𝐵𝐵𝐵�𝐵𝐵

4. Find the measure of angle .

𝑥𝑥

Lesson 7: The Angle Measure of an Arc S.49

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M5 GEOMETRY

5. In the figure, = 126° and = 32°. Find .

𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷

6. In the figure, = 74° and = 42°. is the midpoint of , and is the midpoint of . Find and . 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 𝐾𝐾 𝐶𝐶�𝐶𝐶 𝐽𝐽 𝐵𝐵�𝐵𝐵 𝑚𝑚Solution:∠𝐾𝐾𝐾𝐾𝐾𝐾 Join𝑚𝑚 ∠𝐶𝐶, 𝐶𝐶𝐶𝐶 , , , , and . 𝐵𝐵𝐵𝐵 𝐾𝐾𝐾𝐾 𝐾𝐾𝐾𝐾 𝐾𝐾𝐾𝐾 𝐽𝐽𝐽𝐽 𝐽𝐽𝐽𝐽 =

𝑚𝑚 𝐵𝐵𝐵𝐵� 𝑚𝑚𝐾𝐾�𝐾𝐾 = = 21 2 42˚ 𝑚𝑚 ∠𝐾𝐾𝐾𝐾𝐾𝐾 ˚ =

𝑎𝑎 In , =

△ 𝐵𝐵𝐵𝐵𝐵𝐵 𝑏𝑏 =

𝑐𝑐 =

𝑚𝑚 𝐵𝐵𝐵𝐵� 𝑚𝑚𝐽𝐽�𝐽𝐽 =

𝑚𝑚 ∠𝐽𝐽𝐽𝐽𝐽𝐽 =

𝑑𝑑 = + =

𝑚𝑚 ∠𝐾𝐾𝐾𝐾𝐾𝐾 𝑎𝑎 𝑏𝑏 = + =

𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 𝑐𝑐 𝑑𝑑

Lesson 7: The Angle Measure of an Arc S.50

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M5 GEOMETRY

Lesson 8: Arcs and Chords

Classwork Opening Exercise

Given circle with , = 6, and = 10. Find and . Explain your work.

𝐴𝐴 �𝐵𝐵𝐵𝐵�� ⊥ �𝐷𝐷𝐷𝐷�� 𝐹𝐹𝐹𝐹 𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵 𝐷𝐷𝐷𝐷

Exercises 1. Given circle with = 54° and , find . Explain your work.

𝐴𝐴 𝑚𝑚𝐵𝐵𝐵𝐵� ∠𝐶𝐶𝐶𝐶𝐶𝐶 ≅ ∠𝐷𝐷𝐷𝐷𝐷𝐷 𝑚𝑚𝐷𝐷𝐷𝐷�

Lesson 8: Arcs and Chords S.51

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M5 GEOMETRY

2. If two arcs in a circle have the same measure, what can you say about the quadrilateral formed by the four endpoints? Explain.

3. Find the angle measure of and .

𝐶𝐶𝐶𝐶� 𝐸𝐸𝐸𝐸�

4. = and : : = 1: 2: 4. Find the following angle measures. a. 𝑚𝑚𝐶𝐶𝐶𝐶� 𝑚𝑚𝐸𝐸𝐸𝐸� 𝑚𝑚𝐵𝐵𝐵𝐵� 𝑚𝑚𝐵𝐵�𝐵𝐵 𝑚𝑚𝐸𝐸𝐸𝐸�

𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵

𝑚𝑚∠𝐸𝐸𝐸𝐸𝐸𝐸

𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶

Lesson 8: Arcs and Chords S.52

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M5 GEOMETRY

5. is a diameter of circle . : : = 1: 3: 5. Find the following arc measures. a. 𝐵𝐵𝐵𝐵�� 𝐴𝐴 𝑚𝑚𝐵𝐵�𝐵𝐵 𝑚𝑚𝐷𝐷𝐷𝐷� 𝑚𝑚𝐸𝐸𝐸𝐸�

𝑚𝑚𝐵𝐵�𝐵𝐵

𝑚𝑚𝐷𝐷𝐷𝐷�𝐷𝐷

𝑚𝑚𝐸𝐸𝐸𝐸𝐸𝐸�

Lesson 8: Arcs and Chords S.53

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M5 GEOMETRY

Lesson Summary

Theorems: . Congruent chords have congruent arcs. . Congruent arcs have congruent chords. . Arcs between parallel chords are congruent.

Problem Set

1. Find the following arc measures. a. b. 𝑚𝑚𝐶𝐶𝐶𝐶� c. 𝑚𝑚𝐵𝐵�𝐵𝐵 𝑚𝑚𝐸𝐸𝐸𝐸�

2. In circle , is a diameter, = , and = 32°. a. Find . 𝐴𝐴 �𝐵𝐵𝐵𝐵�� 𝑚𝑚𝐶𝐶𝐶𝐶� 𝑚𝑚𝐸𝐸𝐸𝐸� 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 b. Find . 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴

Lesson 8: Arcs and Chords S.54

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M5 GEOMETRY

3. In circle , is a diameter, 2 = , and . Find .

𝐴𝐴 �𝐵𝐵𝐵𝐵�� 𝑚𝑚𝐶𝐶𝐶𝐶� 𝑚𝑚𝐸𝐸𝐸𝐸� �𝐵𝐵𝐵𝐵�� ∥ �𝐷𝐷𝐷𝐷�� 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶

4. In circle , is a diameter and = 68°.

a. Find𝐴𝐴 �𝐵𝐵𝐵𝐵�� . 𝑚𝑚𝐶𝐶𝐶𝐶� b. Find . 𝑚𝑚𝐶𝐶𝐶𝐶� c. Find 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷

5. In the circle given, . Prove .

𝐵𝐵𝐵𝐵� ≅ 𝐸𝐸𝐸𝐸� �𝐵𝐵��𝐵𝐵� ≅ �𝐷𝐷��𝐷𝐷�

Lesson 8: Arcs and Chords S.55

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M5 GEOMETRY

6. Given circle with , show .

𝐴𝐴 �𝐴𝐴��𝐴𝐴� ∥ �𝐶𝐶𝐶𝐶�� 𝐵𝐵�𝐵𝐵 ≅ 𝐷𝐷𝐷𝐷�

7. In circle , is a radius, , and = 54°. Find . Complete the proof.

𝐴𝐴 �𝐴𝐴��𝐴𝐴� 𝐵𝐵𝐵𝐵� ≅ 𝐵𝐵�𝐵𝐵 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴 =

𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 =

𝑚𝑚∠ 𝑚𝑚∠ + + =

𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 2 +54° = 360°

𝑚𝑚∠ =

𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 =

𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴 =

𝑚𝑚∠ 𝑚𝑚∠ 2 + =

𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 =

𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴

Lesson 8: Arcs and Chords S.56

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M5 GEOMETRY

Lesson 9: Arc Length and Areas of Sectors

Classwork Example 1

a. What is the length of the arc of degree that measures 60° in a circle of radius 10 cm?

b. Given the concentric circles with center and with = 60°, calculate the arc length intercepted by on each circle. The inner circle has a radius of 10, and each circle has a radius 10 units greater than the previous circle. 𝐴𝐴 𝑚𝑚∠𝐴𝐴 ∠𝐴𝐴

c. An arc, again of degree measure 60°, has an arc length of 5 cm. What is the radius of the circle on which the arc sits? 𝜋𝜋

d. Give a general formula for the length of an arc of degree measure ° on a circle of radius .

𝑥𝑥 𝑟𝑟

Lesson 9: Arc Length and Areas of Sectors S.57

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M5 GEOMETRY

e. Is the length of an arc intercepted by an angle proportional to the radius? Explain.

SECTOR: Let be an arc of a circle with center and radius . The union of all segments , where is any point of , is called a sector. 𝐴𝐴𝐴𝐴� 𝑂𝑂 𝑟𝑟 �𝑂𝑂𝑂𝑂�� 𝑃𝑃 𝐴𝐴𝐴𝐴�

Exercise 1 1. The radius of the following circle is 36 cm, and the = 60°.

a. What is the arc length of ? 𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴

𝐴𝐴𝐴𝐴�

b. What is the radian measure of the central angle?

Lesson 9: Arc Length and Areas of Sectors S.58

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M5 GEOMETRY

Example 2

a. Circle has a radius of 10 cm. What is the area of the circle? Write the formula.

𝑂𝑂

b. What is the area of half of the circle? Write and explain the formula.

c. What is the area of a quarter of the circle? Write and explain the formula.

d. Make a conjecture about how to determine the area of a sector defined by an arc measuring 60°.

e. Circle has a minor arc with an angle measure of 60°. Sector has an area of 24 . What is the radius of circle ? 𝑂𝑂 𝐴𝐴𝐴𝐴� 𝐴𝐴𝐴𝐴𝐴𝐴 𝜋𝜋

𝑂𝑂

f. Give a general formula for the area of a sector defined by an arc of angle measure ° on a circle of radius .

𝑥𝑥 𝑟𝑟

Lesson 9: Arc Length and Areas of Sectors S.59

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M5 GEOMETRY

Exercises 2–3 2. The area of sector in the following image is 28 cm . Find the measurement of the central angle labeled °. 2 𝐴𝐴𝐴𝐴𝐴𝐴 𝜋𝜋 𝑥𝑥

3. In the following figure of circle , = 108° and = = 10 cm. 𝑂𝑂 𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴 a. Find . 𝐴𝐴𝐴𝐴� 𝐴𝐴𝐴𝐴�

𝑚𝑚∠𝑂𝑂𝑂𝑂𝑂𝑂

b. Find .

𝑚𝑚𝐵𝐵𝐵𝐵�

c. Find the area of sector .

𝐵𝐵𝐵𝐵𝐵𝐵

Lesson 9: Arc Length and Areas of Sectors S.60

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M5 GEOMETRY

Lesson Summary

Relevant Vocabulary

. ARC: An arc is any of the following three figures—a minor arc, a major arc, or a semicircle. . LENGTH OF AN ARC: The length of an arc is the circular distance around the arc. . MINOR AND MAJOR ARC: In a circle with center , let and be different points that lie on the circle but are not the endpoints of a diameter. The minor arc between and is the set containing , , and all points of the circle that are in the interior of 𝑂𝑂 .𝐴𝐴 The major𝐵𝐵 arc is the set containing , , and all points of the circle that lie in the exterior of . 𝐴𝐴 𝐵𝐵 𝐴𝐴 𝐵𝐵 ∠𝐴𝐴𝐴𝐴𝐴𝐴 𝐴𝐴 𝐵𝐵 . RADIAN: A radian is the measure of the central angle of a sector of a circle with arc length of one radius ∠𝐴𝐴𝐴𝐴𝐴𝐴 length.

. SECTOR: Let be an arc of a circle with center and radius . The union of the segments , where is any point on , is called a sector. is called the arc of the sector, and is called its radius. 𝐴𝐴𝐴𝐴� 𝑂𝑂 𝑟𝑟 𝑂𝑂𝑂𝑂 𝑃𝑃 . SEMICIRCLE: In a circle, let and be the endpoints of a diameter. A semicircle is the set containing , , 𝐴𝐴𝐴𝐴� 𝐴𝐴𝐴𝐴� 𝑟𝑟 and all points of the circle that lie in a given half-plane of the line determined by the diameter. 𝐴𝐴 𝐵𝐵 𝐴𝐴 𝐵𝐵

Problem Set

1. and are points on the circle of radius 5 cm, and the measure of arc is 72°. Find, to one decimal place, each of the following. 𝑃𝑃 𝑄𝑄 𝑃𝑃𝑃𝑃� a. The length of b. The ratio of the arc length to the radius of the circle 𝑃𝑃𝑃𝑃� c. The length of chord d. The distance of the chord from the center of the circle �𝑃𝑃𝑃𝑃�� e. The perimeter of sector �𝑃𝑃𝑃𝑃�� f. The area of the wedge between𝑃𝑃𝑃𝑃𝑃𝑃 the chord and g. The perimeter of this wedge �𝑃𝑃𝑃𝑃�� 𝑃𝑃𝑃𝑃�

2. What is the radius of a circle if the length of a 45° arc is 9 ?

𝜋𝜋 3. and both have an angle measure of 30°, but their arc lengths are not the same. = 4 and = 2. 𝐴𝐴𝐴𝐴� 𝐶𝐶�𝐶𝐶 a. What are the arc lengths of and ? 𝑂𝑂𝑂𝑂 𝐵𝐵𝐵𝐵 b. What is the ratio of the arc length to the radius for both of these 𝐴𝐴𝐴𝐴� 𝐶𝐶�𝐶𝐶 arcs? Explain. c. What are the areas of the sectors and ?

𝐴𝐴𝐴𝐴𝐴𝐴 𝐶𝐶𝐶𝐶𝐶𝐶

Lesson 9: Arc Length and Areas of Sectors S.61

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M5 GEOMETRY

4. In the circles shown, find the value of . Figures are not drawn to scale. a. The circles have central angles of equal measure. b. 𝑥𝑥

𝑥𝑥

c. d.

5. The concentric circles all have center . The measure of the central angle is 45°. The arc lengths are given. 𝐴𝐴 a. Find the radius of each circle. b. Determine the ratio of the arc length to the radius of each circle, and interpret its meaning.

6. In the figure, if the length of is 10 cm, find the length of .

𝑃𝑃𝑃𝑃� 𝑄𝑄𝑄𝑄�

Lesson 9: Arc Length and Areas of Sectors S.62

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M5 GEOMETRY

7. Find, to one decimal place, the areas of the shaded regions. a.

b. The following circle has a radius of 2.

Lesson 9: Arc Length and Areas of Sectors S.63

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M5 GEOMETRY

Lesson 10: Unknown Length and Area Problems

Classwork Opening Exercise In the following figure, a cylinder is carved out from within another cylinder of the same height; the bases of both cylinders share the same center. a. Sketch a cross section of the figure parallel to the base.

b. Mark and label the shorter of the two radii as and the longer of the two radii as . Show how to calculate the area of the shaded region, and explain the parts of the expression. 𝑟𝑟 𝑠𝑠

Exercises 1. Find the area of the following annulus.

Lesson 10: Unknown Length and Area Problems S.64

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M5 GEOMETRY

2. The larger circle of an annulus has a diameter of 10 cm, and the smaller circle has a diameter of 7.6 cm. What is the area of the annulus?

3. In the following annulus, the radius of the larger circle is twice the radius of the smaller circle. If the area of the following annulus is 12 units , what is the radius of the larger circle? 2 𝜋𝜋

4. An ice cream shop wants to design a super straw to serve with its extra thick milkshakes that is double both the width and thickness of a standard straw. A standard straw is 4 mm in diameter and 0.5 mm thick. a. What is the cross-sectional (parallel to the base) area of the new straw (round to the nearest hundredth)?

b. If the new straw is 10 cm long, what is the maximum volume of milkshake that can be in the straw at one time (round to the nearest hundredth)?

Lesson 10: Unknown Length and Area Problems S.65

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M5 GEOMETRY

c. A large milkshake is 32 fl. oz. (approximately 950 mL). If Corbin withdraws the full capacity of a straw 10 times a minute, what is the minimum amount of time that it will take him to drink the milkshake (round to the nearest minute)?

5. In the circle given, is the diameter and is perpendicular to chord . = 8 cm, and = 2 cm. Find , , , the arc length of , and the area of sector (round to the nearest hundredth, if necessary). �𝐸𝐸𝐸𝐸�� 𝐶𝐶𝐶𝐶�� 𝐷𝐷𝐷𝐷 𝐹𝐹𝐹𝐹 𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 𝐶𝐶�𝐶𝐶𝐶𝐶 𝐶𝐶𝐶𝐶𝐶𝐶

6. Given circle with , find the following (round to the nearest hundredth, if necessary). a. 𝐴𝐴 ∠𝐵𝐵𝐵𝐵𝐵𝐵 ≅ ∠𝐵𝐵𝐵𝐵𝐵𝐵

𝑚𝑚𝐶𝐶�𝐶𝐶

𝑚𝑚𝐶𝐶𝐶𝐶�𝐶𝐶

Lesson 10: Unknown Length and Area Problems S.66

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M5 GEOMETRY

𝑚𝑚𝐵𝐵𝐵𝐵𝐵𝐵�

d. Arc length

𝐶𝐶�𝐶𝐶

e. Arc length

𝐶𝐶𝐶𝐶�𝐶𝐶

f. Arc length

𝐵𝐵𝐵𝐵𝐵𝐵�

g. Area of sector

𝐶𝐶𝐶𝐶𝐶𝐶

7. Given circle , find the following (round to the nearest hundredth, if necessary). 𝐴𝐴 a. Circumference of circle

𝐴𝐴

45˚

Lesson 10: Unknown Length and Area Problems S.67

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M5 GEOMETRY

b. Radius of circle

𝐴𝐴

c. Area of sector

𝐶𝐶𝐶𝐶𝐶𝐶

8. Given circle , find the following (round to the nearest hundredth, if necessary). a. 𝐴𝐴

𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶

b. Area of sector

𝐶𝐶𝐶𝐶

Lesson 10: Unknown Length and Area Problems S.68

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M5 GEOMETRY

9. Find the area of the shaded region (round to the nearest hundredth).

10. Many large cities are building or have built mega Ferris wheels. One is 600 feet in diameter and has 48 cars each seating up to 20 people. Each time the Ferris wheel turns degrees, a car is in a position to load. a. How far does a car move with each rotation of degrees (round to the nearest whole number)? 𝜃𝜃

𝜃𝜃

b. What is the value of in degrees?

𝜃𝜃

Lesson 10: Unknown Length and Area Problems S.69

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M5 GEOMETRY

11. is an equilateral triangle with edge length 20 cm. , , and are midpoints of the sides. The vertices of the triangle are the centers of the circles creating the arcs shown. Find the following (round to the nearest hundredth). △ 𝐴𝐴𝐴𝐴𝐴𝐴 𝐷𝐷 𝐸𝐸 𝐹𝐹 a. Area of the sector with center

𝐴𝐴

b. Area of

△ 𝐴𝐴𝐴𝐴𝐴𝐴

c. Area of the shaded region

d. Perimeter of the shaded region

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M5 GEOMETRY

12. In the figure shown, = = 5 cm, = 2 cm, and = 30°. Find the area inside the rectangle but outside of the circles (round to the nearest hundredth). 𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵 𝐺𝐺𝐺𝐺 𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻

13. This is a picture of a piece of a mosaic tile. If the radius of each smaller circle is 1 inch, find the area of the red section, the white section, and the blue section (round to the nearest hundredth).

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M5 GEOMETRY

Problem Set

1. Find the area of the shaded region if the diameter is 32 inches (round to the nearest hundredth).

2. Find the area of the entire circle given the area of the sector.

3. and are arcs of concentric circles with and lying on the radii of the larger circle. Find the area of the region (round to the nearest hundredth). 𝐷𝐷�𝐷𝐷 𝐵𝐵𝐵𝐵� �𝐵𝐵𝐵𝐵�� 𝐹𝐹𝐹𝐹��

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4. Find the radius of the circle, as well as , , and (leave angle measures in radians and arc length in terms of pi). Note that and do not lie on a diameter. 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝐶𝐶 𝐷𝐷

y x

5. In the figure, the radii of two concentric circles are 24 cm and 12 cm. = 120°. If a chord of the larger circle intersects the smaller circle only at , find the area of the shaded region in terms of . 𝑚𝑚𝐷𝐷𝐷𝐷𝐷𝐷� �𝐷𝐷𝐷𝐷�� 𝐶𝐶 𝜋𝜋

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M5 GEOMETRY

Lesson 11: Properties of Tangents

Classwork Exercises 1. and are tangent to circle at points and , respectively. Use a two-column proof to prove = .

�𝐶𝐶𝐶𝐶�� 𝐶𝐶𝐶𝐶�� 𝐴𝐴 𝐷𝐷 𝐸𝐸 𝑎𝑎 𝑏𝑏

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2. In circle , the radius is 9 mm and = 12 mm. a. Find . 𝐴𝐴 𝐵𝐵𝐵𝐵

𝐴𝐴𝐴𝐴

b. Find the area of .

△ 𝐴𝐴𝐴𝐴𝐴𝐴

c. Find the perimeter of quadrilateral .

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴

3. In circle , = 12 and = 13. : = 1: 3. a. Find the radius of the circle. 𝐴𝐴 𝐸𝐸𝐸𝐸 𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴

b. Find (round to the nearest whole number).

𝐵𝐵𝐵𝐵

c. Find .

𝐸𝐸𝐸𝐸

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Lesson Summary

THEOREMS: . A tangent line to a circle is perpendicular to the radius of the circle drawn to the point of tangency. . A line through a point on a circle is tangent at the point if, and only if, it is perpendicular to the radius drawn to the point of tangency.

Relevant Vocabulary

. INTERIOR OF A CIRCLE: The interior of a circle with center and radius is the set of all points in the plane whose distance from the point is less than . 𝑂𝑂 𝑟𝑟 A point in the interior of a circle is said to be inside the circle. A disk is the union of the circle with its 𝑂𝑂 𝑟𝑟 interior. . EXTERIOR OF A CIRCLE: The exterior of a circle with center and radius is the set of all points in the plane whose distance from the point is greater than . 𝑂𝑂 𝑟𝑟 A point exterior to a circle is said to be outside the circle. 𝑂𝑂 𝑟𝑟

. TANGENT TO A CIRCLE: A tangent line to a circle is a line in the same plane that intersects the circle in one and only one point. This point is called the point of tangency. . TANGENT SEGMENT/RAY: A segment is a tangent segment to a circle if the line that contains it is tangent to the circle and one of the end points of the segment is a point of tangency. A ray is called a tangent ray to a circle if the line that contains it is tangent to the circle and the vertex of the ray is the point of tangency. . SECANT TO A CIRCLE: A secant line to a circle is a line that intersects a circle in exactly two points. . POLYGON INSCRIBED IN A CIRCLE: A polygon is inscribed in a circle if all of the vertices of the polygon lie on the circle. . CIRCLE INSCRIBED IN A POLYGON: A circle is inscribed in a polygon if each side of the polygon is tangent to the circle.

Problem Set

1. If = 5, = 12, and = 13, is tangent to circle at point ? Explain. 𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵 𝐴𝐴𝐴𝐴 𝐵𝐵⃖��𝐵𝐵�⃗ 𝐴𝐴 𝐵𝐵

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2. is tangent to circle at point . = 9 and = 15. a. Find the radius of the circle. 𝐵𝐵⃖��𝐵𝐵�⃗ 𝐴𝐴 𝐵𝐵 𝐷𝐷𝐷𝐷 𝐵𝐵𝐵𝐵 b. Find .

𝐴𝐴𝐴𝐴

3. A circular pond is fenced on two opposite sides ( , ) with wood and the other two sides with metal fencing. If all four sides of fencing are tangent to the pond, is there more wood or metal fencing used? �𝐶𝐶𝐶𝐶�� 𝐹𝐹𝐹𝐹��

4. Find if the line shown is tangent to the circle at point .

𝑥𝑥 𝐵𝐵

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5. is tangent to the circle at point , and = .

⃖𝑃𝑃a.��𝑃𝑃�⃗ Find ( ). 𝐶𝐶 𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷 b. Find ( ). 𝑥𝑥 𝑚𝑚𝐶𝐶𝐶𝐶�

𝑦𝑦 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶

6. Construct two lines tangent to circle through point .

𝐴𝐴 𝐵𝐵

7. Find , the length of the common tangent line between the two circles (round to the nearest hundredth).

𝑥𝑥

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8. is tangent to both circles and . The radius of circle is 9, and the radius of circle is 5. The circles are 2 units apart. Find the length of , or (round to the nearest hundredth). 𝐸𝐸��𝐸𝐸� 𝐴𝐴 𝐶𝐶 𝐴𝐴 𝐶𝐶

�𝐸𝐸��𝐸𝐸� 𝑥𝑥

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 M5 GEOMETRY

Lesson 12: Tangent Segments

Classwork Opening Exercise In the diagram, what do you think the length of could be? How do you know?

𝑧𝑧

Example

In each diagram, try to draw a circle with center that is tangent to both rays of .

a. 𝐷𝐷 b. ∠𝐵𝐵𝐵𝐵𝐵𝐵

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 M5 GEOMETRY

In which diagrams did it seem impossible to draw such a circle? Why did it seem impossible?

What do you conjecture about circles tangent to both rays of an angle? Why do you think that?

Exercises 1. You conjectured that if a circle is tangent to both rays of a circle, then the center lies on the angle bisector. a. Rewrite this conjecture in terms of the notation suggested by the diagram. Given:

Need to show:

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 M5 GEOMETRY

b. Prove your conjecture using a two-column proof.

2. An angle is shown below. a. Draw at least three different circles that are tangent to both rays of the given angle.

b. Label the center of one of your circles with . How does the distance between and the rays of the angle compare to the radius of the circle? How do you know? 𝑃𝑃 𝑃𝑃

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3. Construct as many circles as you can that are tangent to both the given angles at the same time. You can extend the rays as needed. These two angles share a side.

Explain how many circles you can draw to meet the above conditions and how you know.

4. In a triangle, let be the location where two angle bisectors meet. Must be on the third angle bisector as well? Explain your reasoning. 𝑃𝑃 𝑃𝑃

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 M5 GEOMETRY

5. Using a straightedge, draw a large triangle .

𝐴𝐴𝐴𝐴𝐴𝐴

a. Construct a circle inscribed in the given triangle.

b. Explain why your construction works.

c. Do you know another name for the intersection of the angle bisectors in relation to the triangle?

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 M5 GEOMETRY

Lesson Summary

THEOREMS: . The two tangent segments to a circle from an exterior point are congruent. . If a circle is tangent to both rays of an angle, then its center lies on the angle bisector. . Every triangle contains an inscribed circle whose center is the intersection of the triangle’s angle bisectors.

Problem Set

1. On a piece of paper, draw a circle with center and a point, , outside of the circle. a. How many tangents can you draw from to the circle? 𝐴𝐴 𝐶𝐶 b. Draw two tangents from to the circle, and label the tangency points and . Fold your paper along the line 𝐶𝐶 . What do you notice about the lengths of and ? About the measures of and ? 𝐶𝐶 𝐷𝐷 𝐸𝐸 c. is the of . 𝐴𝐴𝐴𝐴 �𝐶𝐶𝐶𝐶�� 𝐶𝐶𝐶𝐶�� ∠𝐷𝐷𝐷𝐷𝐷𝐷 ∠𝐸𝐸𝐸𝐸𝐸𝐸 d. and are tangent to circle . Find . 𝐴𝐴𝐴𝐴�� ∠𝐷𝐷𝐷𝐷𝐷𝐷

�𝐶𝐶𝐶𝐶�� 𝐶𝐶𝐶𝐶�� 𝐴𝐴 𝐴𝐴𝐴𝐴 2 = 2. In the figure, the three segments are tangent to the circle at points , , and . If 3 , find , , and . 𝐵𝐵 𝐹𝐹 𝐺𝐺 𝑦𝑦 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑧𝑧

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 M5 GEOMETRY

3. In the figure given, the three segments are tangent to the circle at points , , and . a. Prove = + . 𝐽𝐽 𝐼𝐼 𝐻𝐻 b. Find the perimeter of . 𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺 𝐻𝐻𝐻𝐻

△ 𝐺𝐺𝐺𝐺𝐺𝐺

4. In the figure given, the three segments are tangent to the circle at points F, , and . Find .

𝐵𝐵 𝐺𝐺 𝐷𝐷𝐷𝐷

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 M5 GEOMETRY

5. is tangent to circle . If points and are the intersection points of circle and any line parallel to , answer the following. 𝐸𝐸⃖��𝐸𝐸�⃗ 𝐴𝐴 𝐶𝐶 𝐷𝐷 𝐴𝐴 ⃖𝐸𝐸��𝐸𝐸�⃗

a. Does = for any line parallel to ? Explain.

b. Suppose𝐶𝐶𝐶𝐶 that𝐺𝐺 𝐺𝐺 coincides with . Would⃖𝐸𝐸��𝐸𝐸�⃗ , , and all coincide with ? c. Suppose , ,⃖𝐶𝐶𝐶𝐶 �and��⃗ have now reached⃖𝐸𝐸��𝐸𝐸�⃗ , so𝐶𝐶 𝐺𝐺 is tangent𝐷𝐷 to the circle. 𝐵𝐵What is the angle between and ? 𝐶𝐶 𝐺𝐺 𝐷𝐷 𝐵𝐵 ⃖𝐶𝐶𝐶𝐶��⃗ ⃖𝐶𝐶𝐶𝐶��⃗ d. Draw another line tangent to the circle from some point, , in the exterior of the circle. If the point of �𝐴𝐴𝐴𝐴�� tangency is point , what is the measure of ? 𝑃𝑃

𝑇𝑇 ∠𝑃𝑃𝑃𝑃𝑃𝑃 6. The segments are tangent to circle at points and . is a diameter of the circle. a. Prove . 𝐴𝐴 𝐵𝐵 𝐷𝐷 �𝐸𝐸��𝐸𝐸� b. Prove quadrilateral is a kite. 𝐵𝐵��𝐵𝐵� ‖ �𝐶𝐶𝐶𝐶��

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴

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7. In the diagram shown, is tangent to the circle at point . What is the relationship between , the angle between the tangent and a chord, and the arc subtended by that chord and its inscribed angle ? ⃖𝐵𝐵��𝐵𝐵�⃗ 𝐵𝐵 ∠𝐷𝐷𝐷𝐷𝐷𝐷

∠𝐷𝐷𝐷𝐷𝐷𝐷

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 13 M5 GEOMETRY

Lesson 13: The Inscribed Angle Alternate—A Tangent Angle

Classwork Opening Exercise In circle , = 56°, and is a diameter. Find the listed measure, and explain your answer.

a. 𝐴𝐴 𝑚𝑚𝐵𝐵𝐵𝐵� �𝐵𝐵𝐵𝐵��

𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵

𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷

𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵

𝑚𝑚𝐵𝐵𝐵𝐵�

𝑚𝑚𝐷𝐷𝐷𝐷�

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 13 M5 GEOMETRY

g. Does measure 56°? Explain.

∠𝐵𝐵𝐵𝐵𝐵𝐵

h. How do you think we could determine the measure of ?

∠𝐵𝐵𝐵𝐵𝐵𝐵

Exploratory Challenge

Diagram 1 Diagram 2

Examine the diagrams shown. Develop a conjecture about the relationship between and .

𝑎𝑎 𝑏𝑏

Test your conjecture by using a protractor to measure and .

𝑎𝑎 𝑏𝑏

Diagram 1 𝒂𝒂 𝒃𝒃

Diagram 2

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 13 M5 GEOMETRY

Do your measurements confirm the relationship you found in your homework? If needed, revise your conjecture about the relationship between and :

𝑎𝑎 𝑏𝑏

Now, test your conjecture further using the circle below.

𝒂𝒂 𝒃𝒃

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 13 M5 GEOMETRY

Now, we will prove your conjecture, which is stated below as a theorem.

THE TANGENT-SECANT THEOREM: Let be a point on a circle, let be a tangent ray to the circle, and let be a point on the circle such that is a secant to the circle. If = and is the angle measure of the arc intercepted by , ��⃗ 1 𝐴𝐴 𝐴𝐴𝐴𝐴 𝐶𝐶 = then 2 . ⃖𝐴𝐴��𝐴𝐴�⃗ 𝑎𝑎 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 𝑏𝑏 ∠𝐵𝐵𝐵𝐵𝐵𝐵 𝑎𝑎 𝑏𝑏

Given circle with tangent , prove what we have just discovered using what you know about the properties of a circle and tangent and secant ⃖��⃗ lines. 𝑂𝑂 𝐴𝐴𝐴𝐴 a. Draw triangle . What is the measure of ? Explain.

𝐴𝐴𝐴𝐴𝐴𝐴 ∠𝐴𝐴𝐴𝐴𝐴𝐴

b. What is the measure of ? Explain.

∠𝑂𝑂𝑂𝑂𝑂𝑂

c. Express the measure of the remaining two angles of triangle in terms of and explain.

𝐴𝐴𝐴𝐴𝐴𝐴 𝑎𝑎

d. What is the measure of in terms of ? Show how you got the answer.

∠𝐴𝐴𝐴𝐴𝐴𝐴 𝑎𝑎

e. Explain to your neighbor what we have just proven.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 13 M5 GEOMETRY

Exercises Find , , , , and/or .

1. 𝑥𝑥 𝑦𝑦 𝑎𝑎 𝑏𝑏 𝑐𝑐

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 13 M5 GEOMETRY

𝑎𝑎

𝑏𝑏

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Lesson Summary

THEOREMS:

. CONJECTURE: Let be a point on a circle, let be a tangent ray to the circle, and let be a point on the circle such that is a secant to the circle. If = and b is the angle measure of the arc ��⃗ 𝐴𝐴 1 𝐴𝐴𝐴𝐴 𝐶𝐶 = intercepted by ⃖𝐴𝐴��𝐴𝐴�⃗ , then 2 . 𝑎𝑎 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 . THE TANGENT-SECANT∠𝐵𝐵𝐵𝐵 THEOREM𝐵𝐵 :𝑎𝑎 Let 𝑏𝑏 be a point on a circle, let be a tangent ray to the circle, and let be a point on the circle such that is a secant to the circle. If = and is the angle measure of ��⃗ 𝐴𝐴 1 𝐴𝐴𝐴𝐴 𝐶𝐶 = the arc intercepted by , then⃖𝐴𝐴��𝐴𝐴�⃗ 2 . 𝑎𝑎 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 𝑏𝑏 . Suppose is a chord ∠of𝐵𝐵 circle𝐵𝐵𝐵𝐵 , and𝑎𝑎 𝑏𝑏 is a tangent segment to the circle at point . If is any point other than or in the arc of on the opposite side of from , then = . �𝐴𝐴��𝐴𝐴� 𝐶𝐶 𝐴𝐴��𝐴𝐴� 𝐴𝐴 𝐸𝐸

𝐴𝐴 𝐵𝐵 𝐶𝐶 �𝐴𝐴��𝐴𝐴� 𝐷𝐷 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵

Problem Set

In Problems 1–9, solve for , , and/or .

1. 𝑎𝑎 𝑏𝑏 𝑐𝑐 2. 3.

4. 5. 6.

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7. 8. 9.

10. is tangent to circle . is a diameter. Find the angle measurements. a. ⃖𝐵𝐵𝐵𝐵��⃗ 𝐴𝐴 �𝐷𝐷𝐷𝐷�� b. 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 c. 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 d. 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 e. 𝑚𝑚∠𝐹𝐹𝐹𝐹𝐹𝐹

𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵

11. is tangent to circle . is a diameter. Prove: (i) = and (ii) = .

𝐵𝐵𝐵𝐵�� 𝐴𝐴 �𝐵𝐵��𝐵𝐵� 𝑓𝑓 𝑎𝑎 𝑑𝑑 𝑐𝑐

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Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle

Classwork Opening Exercise

is tangent to the circle as shown. a. Find the values of and . ⃖𝐷𝐷𝐷𝐷��⃗

𝑎𝑎 𝑏𝑏

b. Is a diameter of the circle? Explain.

𝐶𝐶��𝐶𝐶�

Exercises 1–2 1. In circle , is a radius, and = 142°. Find , and explain how you know.

𝑃𝑃 �𝑃𝑃𝑃𝑃�� 𝑚𝑚𝑀𝑀�𝑀𝑀 𝑚𝑚∠𝑀𝑀𝑀𝑀𝑀𝑀

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2. In the circle shown, = 55°. Find and . Explain your answer.

𝑚𝑚𝐶𝐶𝐶𝐶� 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 𝑚𝑚𝐸𝐸�𝐸𝐸

Example

a. Find . Justify your answer.

𝑥𝑥

b. Find .

𝑥𝑥

Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle S.99

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 M5 GEOMETRY

We can state the results of part (b) of this example as the following theorem:

SECANT ANGLE THEOREM—INTERIOR CASE: The measure of an angle whose vertex lies in the interior of a circle is equal to half the sum of the angle measures of the arcs intercepted by it and its vertical angle.

Exercises 3–7 In Exercises 3–5, find and .

3. 𝑥𝑥 𝑦𝑦 4.

Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle S.100

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 M5 GEOMETRY

6. In the circle shown, is a diameter. Find and .

�𝐵𝐵𝐵𝐵�� 𝑥𝑥 𝑦𝑦

3 : = 2: 1 = 180 7. In the circle shown, is a diameter. . Prove 2 using a two-column proof. �𝐵𝐵𝐵𝐵�� 𝐷𝐷𝐷𝐷 𝐵𝐵𝐵𝐵 𝑦𝑦 − 𝑥𝑥

Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle S.101

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 M5 GEOMETRY

Lesson Summary

THEOREM:

. SECANT ANGLE THEOREM—INTERIOR CASE: The measure of an angle whose vertex lies in the interior of a circle is equal to half the sum of the angle measures of the arcs intercepted by it and its vertical angle.

Relevant Vocabulary

Problem Set

In Problems 1–4, find .

1. 𝑥𝑥 2.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 M5 GEOMETRY

3. 4.

5. Find ( ) and ( ).

𝑥𝑥 𝑚𝑚𝐶𝐶𝐶𝐶� 𝑦𝑦 𝑚𝑚𝐷𝐷�𝐷𝐷

6. Find the ratio of : .

𝑚𝑚𝐸𝐸𝐸𝐸� 𝑚𝑚𝐷𝐷𝐷𝐷�

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 M5 GEOMETRY

7. is a diameter of circle . Find .

𝐵𝐵𝐵𝐵�� 𝐴𝐴 𝑥𝑥

8. Show that the general formula we discovered in Example 1 also works for central angles. (Hint: Extend the radii to form two diameters, and use relationships between central angles and arc measure.)

Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle S.104

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M5 GEOMETRY

Lesson 15: Secant Angle Theorem, Exterior Case

Classwork Opening Exercise 1. Shown below are circles with two intersecting secant chords.

Measure , , and in the two diagrams. Make a conjecture about the relationship between them.

𝑎𝑎 𝑏𝑏 𝑐𝑐

𝒂𝒂 𝒃𝒃 𝒄𝒄

CONJECTURE about the relationship between , , and :

𝑎𝑎 𝑏𝑏 𝑐𝑐

Lesson 15: Secant Angle Theorem, Exterior Case S.105

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M5 GEOMETRY

2. We will prove the following.

We can interpret this statement in terms of the diagram below. Let and be the angle measures of the arcs + = intercepted by and . Then measure is the average of 𝑏𝑏 and 𝑐𝑐; that is, 2 . 𝑏𝑏 𝑐𝑐 ∠𝑆𝑆𝑆𝑆𝑆𝑆 ∠𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎

a. Find as many pairs of congruent angles as you can in the diagram below. Express the measures of the angles in terms of and whenever possible.

𝑏𝑏 𝑐𝑐

b. Which triangles in the diagram are similar? Explain how you know.

c. See if you can use one of the triangles to prove the secant angle theorem, interior case. (Hint: Use the exterior angle theorem.)

Lesson 15: Secant Angle Theorem, Exterior Case S.106

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M5 GEOMETRY

Exploratory Challenge Shown below are two circles with two secant chords intersecting outside the circle.

Measure , , and . Make a conjecture about the relationship between them.

𝑎𝑎 𝑏𝑏 𝑐𝑐

𝒂𝒂 𝒃𝒃 𝒄𝒄

Conjecture about the relationship between , , and :

𝑎𝑎 𝑏𝑏 𝑐𝑐

Test your conjecture with another diagram.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M5 GEOMETRY

Exercises Find , , and/or .

1. 𝑥𝑥 𝑦𝑦 𝑧𝑧 2.

3. 4.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M5 GEOMETRY

Closing Exercise We have just developed proofs for an entire family of theorems. Each theorem in this family deals with two shapes and how they overlap. The two shapes are two intersecting lines and a circle. In this exercise, you will summarize the different cases.

The Inscribed Angle Theorem and its Family of Theorems How the Two Shapes Relationship between Diagram Overlap , , , and

𝒂𝒂 𝒃𝒃 𝒄𝒄 𝒅𝒅

(Inscribed Angle Theorem)

(Secant–Tangent)

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M5 GEOMETRY

(Secant Angle Theorem—Interior)

(Secant Angle Theorem—Exterior)

(Two Tangent Lines)

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M5 GEOMETRY

Lesson Summary

THEOREMS:

. SECANT ANGLE THEOREM—EXTERIOR CASE: The measure of an angle whose vertex lies in the exterior of the circle, and each of whose sides intersect the circle in two points, is equal to half the difference of the angle measures of its larger and smaller intercepted arcs.

Relevant Vocabulary

SECANT TO A CIRCLE: A secant line to a circle is a line that intersects a circle in exactly two points.

Problem Set

1. Find . 2. Find and .

𝑥𝑥 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷

Lesson 15: Secant Angle Theorem, Exterior Case S.111

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M5 GEOMETRY

3. Find , , and . 4. Find and .

𝑚𝑚∠𝐸𝐸𝐸𝐸𝐸𝐸 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 𝑚𝑚∠𝐹𝐹𝐹𝐹𝐹𝐹 𝑚𝑚∠𝐹𝐹𝐹𝐹𝐹𝐹

5. Find and .

𝑥𝑥 𝑦𝑦

6. The radius of circle is 4. and are tangent to the circle with = 12. Find and the area of quadrilateral rounded to the nearest hundredth. 𝐴𝐴 �𝐷𝐷��𝐷𝐷� 𝐶𝐶𝐶𝐶�� 𝐷𝐷𝐷𝐷 𝑚𝑚𝐷𝐷𝐷𝐷�

𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M5 GEOMETRY

7. Find the measure of , , and . 8. Find the values of and .

𝐵𝐵𝐵𝐵� 𝐹𝐹𝐹𝐹� 𝐺𝐺𝐺𝐺� 𝑥𝑥 𝑦𝑦

9. The radius of a circle is 6. a. If the angle formed between two tangent lines to the circle is 60°, how long are the segments between the point of intersection of the tangent lines and the circle? b. If the angle formed between the two tangent lines is 120°, how long are each of the segments between the point of intersection of the tangent lines and the point of tangency? Round to the nearest hundredth.

10. and are tangent to circle . Prove = .

�𝐷𝐷��𝐷𝐷� �𝐸𝐸��𝐸𝐸� 𝐴𝐴 𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵

Lesson 15: Secant Angle Theorem, Exterior Case S.113

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 16 M5 GEOMETRY

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant- Tangent) Diagrams

Classwork Opening Exercise Identify the type of angle and the angle/arc relationship, and then find the measure of .

a. b. 𝑥𝑥

c. d.

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams S.114

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 16 M5 GEOMETRY

Exploratory Challenge 1 Measure the lengths of the chords in centimeters, and record them in the table. A. B.

C. D.

Circle ( ) ( ) ( ) ( ) Do you notice a relationship?

A 𝒂𝒂 𝐜𝐜𝐜𝐜 𝒃𝒃 𝐜𝐜𝐜𝐜 𝒄𝒄 𝐜𝐜𝐜𝐜 𝒅𝒅 𝐜𝐜𝐜𝐜

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams S.115

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 16 M5 GEOMETRY

Exploratory Challenge 2 Measure the lengths of the chords in centimeters, and record them in the table. A. B.

C. D.

Circle ( ) ( ) ( ) ( ) Do you notice a relationship?

A 𝒂𝒂 𝐜𝐜𝐜𝐜 𝒃𝒃 𝐜𝐜𝐜𝐜 𝒄𝒄 𝐜𝐜𝐜𝐜 𝒅𝒅 𝐜𝐜𝐜𝐜

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams S.116

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 16 M5 GEOMETRY

The Inscribed Angle Theorem and Its Family

Relationship between Diagram How the two shapes overlap , , , and

𝒂𝒂 𝒃𝒃 𝒄𝒄 𝒅𝒅

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams S.117

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 16 M5 GEOMETRY

Lesson Summary

THEOREMS: . When secant lines intersect inside a circle, use = .

. When secant lines intersect outside of a circle, 𝑎𝑎use∙ 𝑏𝑏 ( 𝑐𝑐+∙ 𝑑𝑑 ) = ( + ). . When a tangent line and a secant line intersect outside𝑎𝑎 𝑎𝑎 of 𝑏𝑏a circle,𝑐𝑐 𝑐𝑐 use𝑑𝑑 = ( + ). 2 𝑎𝑎 𝑏𝑏 𝑏𝑏 𝑐𝑐 Relevant Vocabulary

SECANT TO A CIRCLE: A secant line to a circle is a line that intersects a circle in exactly two points.

Problem Set

1. Find . 2. Find .

𝑥𝑥 𝑥𝑥

3. < , 1, < , and all values are 4. = 6, = 9, and = 18. Show = 3. integers; prove = 3. 𝐷𝐷𝐷𝐷 𝐹𝐹𝐹𝐹 𝐷𝐷𝐷𝐷 ≠ 𝐷𝐷𝐷𝐷 𝐹𝐹𝐹𝐹 𝐶𝐶𝐶𝐶 𝐶𝐶𝐶𝐶 𝐶𝐶𝐶𝐶 𝐶𝐶𝐶𝐶

𝐷𝐷𝐷𝐷

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams S.118

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 16 M5 GEOMETRY

5. Find . 6. Find .

𝑥𝑥 𝑥𝑥

7. Find . 8. Find .

𝑥𝑥 𝑥𝑥

9. In the circle shown, = 11, = 10, and = 8. Find , , .

𝐷𝐷𝐷𝐷 𝐵𝐵𝐵𝐵 𝐷𝐷𝐷𝐷 𝐹𝐹𝐹𝐹 𝐵𝐵𝐵𝐵 𝐹𝐹𝐹𝐹

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams S.119

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 16 M5 GEOMETRY

10. In the circle shown, = 150°, = 30°, = 60°, = 8, = 4, and = 12. a. Find . 𝑚𝑚𝐷𝐷�𝐷𝐷𝐷𝐷 𝑚𝑚𝐷𝐷�𝐷𝐷 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷 𝐷𝐷𝐷𝐷 𝐺𝐺𝐺𝐺 b. Prove ~ . 𝑚𝑚∠𝐺𝐺𝐺𝐺𝐺𝐺 c. Set up a proportion using and . △ 𝐷𝐷𝐷𝐷𝐷𝐷 △ 𝐸𝐸𝐸𝐸𝐸𝐸 d. Set up an equation with and using a theorem 𝐶𝐶��𝐶𝐶� 𝐺𝐺𝐺𝐺�� for segment lengths from this section. 𝐶𝐶𝐶𝐶 𝐺𝐺𝐺𝐺

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams S.120

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M5 GEOMETRY

Lesson 17: Writing the Equation for a Circle

Classwork Exercises 1–2 1. What is the length of the segment shown on the coordinate plane below?

2. Use the distance formula to determine the distance between points (9, 15) and (3, 7).

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M5 GEOMETRY

Example 1

If we graph all of the points whose distance from the origin is equal to 5, what shape will be formed?

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M5 GEOMETRY

Example 2

Let’s look at another circle, one whose center is not at the origin. Shown below is a circle with center (2, 3) and radius 5.

Lesson 17: Writing the Equation for a Circle S.123

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M5 GEOMETRY

Exercises 3–11 3. Write an equation for the circle whose center is at (9, 0) and has radius 7.

4. Write an equation whose graph is the circle below.

5. What is the radius and center of the circle given by the equation ( + 12) + ( 4) = 81? 2 2 𝑥𝑥 𝑦𝑦 −

6. Petra is given the equation ( 15) + ( + 4) = 100 and identifies its graph as a circle whose center is ( 15, 4) and radius is 10. Has Petra made a mistake?2 Explain.2 𝑥𝑥 − 𝑦𝑦 −

Lesson 17: Writing the Equation for a Circle S.124

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M5 GEOMETRY

7. a. What is the radius of the circle with center (3, 10) that passes through (12, 12)?

b. What is the equation of this circle?

8. A circle with center (2, 5) is tangent to the -axis. a. What is the radius of the circle? − 𝑥𝑥

b. What is the equation of the circle?

9. Two points in the plane, ( 3, 8) and (17, 8), represent the endpoints of the diameter of a circle. a. What is the center of the circle? Explain. 𝐴𝐴 − 𝐵𝐵

b. What is the radius of the circle? Explain.

Lesson 17: Writing the Equation for a Circle S.125

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M5 GEOMETRY

c. Write the equation of the circle.

10. Consider the circles with the following equations: + = 25 and 2 2 ( 9𝑥𝑥) +𝑦𝑦( 12) = 100. 2 2 a. What are the radii of the circles? 𝑥𝑥 − 𝑦𝑦 −

b. What is the distance between the centers of the circles?

c. Make a rough sketch of the two circles to explain why the circles must be tangent to one another.

Lesson 17: Writing the Equation for a Circle S.126

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M5 GEOMETRY

11. A circle is given by the equation ( + 2 + 1) + ( + 4 + 4) = 121. a. What is the center of the circle?2 2 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑦𝑦

b. What is the radius of the circle?

c. Describe what you had to do in order to determine the center and the radius of the circle.

Lesson 17: Writing the Equation for a Circle S.127

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M5 GEOMETRY

Lesson Summary

( ) + ( ) = is the center-radius form of the general equation for any circle with radius and center ( , ). 2 2 2 𝑥𝑥 − 𝑎𝑎 𝑦𝑦 − 𝑏𝑏 𝑟𝑟 𝑟𝑟

𝑎𝑎 𝑏𝑏

Problem Set

1. Write the equation for a circle with center , and radius 13. 1 3

�2 7� √ 2. What is the center and radius of the circle given by the equation + ( 11) = 144? 2 2 𝑥𝑥 𝑦𝑦 − 3. A circle is given by the equation + = 100. Which of the following points are on the circle? a. (0, 10) 2 2 𝑥𝑥 𝑦𝑦 b. ( 8, 6) c. ( 10, 10) − d. (45, 55) − − e. ( 10, 0)

− 4. Determine the center and radius of each circle. a. 3 + 3 = 75 b. 2( 2 + 1) 2+ 2( + 2) = 10 𝑥𝑥 𝑦𝑦 c. 4( 2)2 + 4( 9)2 64 = 0 𝑥𝑥 𝑦𝑦 2 2 𝑥𝑥 − 𝑦𝑦 − − 5. A circle has center ( 13, ) and passes through the point (2, ). a. What is the radius of the circle? − 𝜋𝜋 𝜋𝜋 b. Write the equation of the circle.

6. Two points in the plane, (19, 4) and (19, 6), represent the endpoints of the diameter of a circle. a. What is the center of the circle? 𝐴𝐴 𝐵𝐵 − b. What is the radius of the circle? c. Write the equation of the circle.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M5 GEOMETRY

7. Write the equation of the circle shown below.

8. Write the equation of the circle shown below.

9. Consider the circles with the following equations: + = 2 and 2 2 ( 𝑥𝑥3) +𝑦𝑦( 3) = 32. 2 2 a. What are the radii of the two circles? 𝑥𝑥 − 𝑦𝑦 − b. What is the distance between their centers? c. Make a rough sketch of the two circles to explain why the circles must be tangent to one another.

Lesson 17: Writing the Equation for a Circle S.129

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M5 GEOMETRY

Lesson 18: Recognizing Equations of Circles

Classwork Opening Exercise a. Express this as a trinomial: ( 5) . 2 3 𝑥𝑥 − 3 𝑥𝑥 2 𝑥𝑥 𝑥𝑥 𝑥𝑥 40 3 3 ? =

b. Express this as a trinomial: ( + 4) . 𝑥𝑥 2 𝑥𝑥 3 3 𝑥𝑥 2 𝑥𝑥 𝑥𝑥 𝑥𝑥 49 3 3 9 = c. Factor the trinomial: + 12 + 36. 2 𝑥𝑥 𝑥𝑥 𝑥𝑥

d. Complete the square to solve the following equation: + 6 = 40. 2 𝑥𝑥 𝑥𝑥

Lesson 18: Recognizing Equations of Circles S.130

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M5 GEOMETRY

Example 1

The following is the equation of a circle with radius 5 and center (1, 2). Do you see why? 2 + 1 + 4 + 4 = 25 2 2 𝑥𝑥 − 𝑥𝑥 𝑦𝑦 − 𝑦𝑦

Exercise 1 1. Rewrite the following equations in the form ( ) + ( ) = . a. + 4 + 4 + 6 + 9 = 36 2 2 2 𝑥𝑥 − 𝑎𝑎 𝑦𝑦 − 𝑏𝑏 𝑟𝑟 2 2 𝑥𝑥 𝑥𝑥 𝑦𝑦 − 𝑥𝑥

b. 10 + 25 + + 14 + 49 = 4 2 2 𝑥𝑥 − 𝑥𝑥 𝑦𝑦 𝑦𝑦

Lesson 18: Recognizing Equations of Circles S.131

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M5 GEOMETRY

Example 2

What is the center and radius of the following circle? + 4 + 12 = 41 2 2 𝑥𝑥 𝑥𝑥 𝑦𝑦 − 𝑦𝑦

Exercises 2–4 2. Identify the center and radius for each of the following circles. a. 20 + + 6 = 35 2 2 𝑥𝑥 − 𝑥𝑥 𝑦𝑦 𝑦𝑦

19 3 + 5 = b. 2 2 2 𝑥𝑥 − 𝑥𝑥 𝑦𝑦 − 𝑦𝑦

Lesson 18: Recognizing Equations of Circles S.132

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M5 GEOMETRY

3. Could the circle with equation 6 + 7 = 0 have a radius of 4? Why or why not? 2 2 𝑥𝑥 − 𝑥𝑥 𝑦𝑦 −

4. Stella says the equation 8 + + 2y = 5 has a center of (4, 1) and a radius of 5. Is she correct? Why or why not? 2 2 𝑥𝑥 − 𝑥𝑥 𝑦𝑦 −

Example 3

Could + + + + = 0 represent a circle? 2 2 𝑥𝑥 𝑦𝑦 𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵 𝐶𝐶

Lesson 18: Recognizing Equations of Circles S.133

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M5 GEOMETRY

Exercise 5 5. Identify the graphs of the following equations as a circle, a point, or an empty set. a. + + 4 = 0 2 2 𝑥𝑥 𝑦𝑦 𝑥𝑥

b. + + 6 4 + 15 = 0 2 2 𝑥𝑥 𝑦𝑦 𝑥𝑥 − 𝑦𝑦

13 2 + 2 5 + + = 0 c. 4 2 2 𝑥𝑥 𝑦𝑦 − 𝑥𝑥 𝑦𝑦

Lesson 18: Recognizing Equations of Circles S.134

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M5 GEOMETRY

Problem Set

1. Identify the centers and radii of the following circles. a. ( + 25) + = 1 b. + 2 +2 2 8 = 8 𝑥𝑥 𝑦𝑦 c. 2 20 + 2 10 + 25 = 0 𝑥𝑥 𝑥𝑥 𝑦𝑦 − 𝑦𝑦 d. 2 + = 19 2 𝑥𝑥 − 𝑥𝑥 𝑦𝑦 − 𝑦𝑦 2 2 1 + + + = e. 𝑥𝑥 𝑦𝑦 4 2 2 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑦𝑦 − 2. Sketch graphs of the following equations. a. + + 10 4 + 33 = 0 b. 2 + 2 + 14 16 + 104 = 0 𝑥𝑥 𝑦𝑦 𝑥𝑥 − 𝑦𝑦 c. 2 + 2 + 4 10 + 29 = 0 𝑥𝑥 𝑦𝑦 𝑥𝑥 − 𝑦𝑦 2 2 𝑥𝑥 𝑦𝑦 𝑥𝑥 − 𝑦𝑦 3. Chante claims that two circles given by ( + 2) + ( 4) = 49 and + 6 + 16 + 37 = 0 are externally tangent. She is right. Show that she is. 2 2 2 2 𝑥𝑥 𝑦𝑦 − 𝑥𝑥 𝑦𝑦 − 𝑥𝑥 𝑦𝑦

4. Draw a circle. Randomly select a point in the interior of the circle; label the point . Construct the greatest radius circle with center that lies within the circular region defined by the original circle. Hint: Draw a line through the center, the circle, and point . 𝐴𝐴 𝐴𝐴 𝐴𝐴

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M5 GEOMETRY

Lesson 19: Equations for Tangent Lines to Circles

Classwork Opening Exercise A circle of radius 5 passes through points ( 3, 3) and (3, 1).

a. What is the special name for segment𝐴𝐴 − ? 𝐵𝐵 𝐴𝐴𝐴𝐴

b. How many circles can be drawn that meet the given criteria? Explain how you know.

c. What is the slope of ?

�𝐴𝐴��𝐴𝐴�

d. Find the midpoint of .

�𝐴𝐴��𝐴𝐴�

e. Find the equation of the line containing a diameter of the given circle perpendicular to .

�𝐴𝐴��𝐴𝐴�

f. Is there more than one answer possible for part (e)?

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M5 GEOMETRY

Example 1

Consider the circle with equation ( 3) + ( 5) = 20. Find the equations of two tangent lines to the circle that 1 2 2 each has slope 2. 𝑥𝑥 − 𝑦𝑦 − −

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M5 GEOMETRY

Exercise 1 Consider the circle with equation ( 4) + ( 5) = 20. Find the equations of two tangent lines to the circle that each has slope 2. 2 2 𝑥𝑥 − 𝑦𝑦 −

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Example 2

Refer to the diagram below. Let > 1. What is the equation of the tangent line to the circle + = 1 through the point ( , 0) on the -axis with a point of tangency in the upper half-plane? 2 2 𝑝𝑝 𝑥𝑥 𝑦𝑦 𝑝𝑝 𝑥𝑥

Exercises 2–4 2. Use the same diagram from Example 2 above, but label the point of tangency in the lower half-plane as . a. What are the coordinates of ? 𝑄𝑄′ ′ 𝑄𝑄

b. What is the slope of ?

�𝑂𝑂��𝑂𝑂��′

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M5 GEOMETRY

c. What is the slope of ?

�𝑄𝑄��′�𝑃𝑃�

d. Find the equation of the second tangent line to the circle through ( , 0).

𝑝𝑝

3. Show that a circle with equation ( 2) + ( + 3) = 160 has two tangent lines with equations 1 1 2 2 + 15 = ( 6) 9 = ( + 2) 3 and 3𝑥𝑥 − . 𝑦𝑦 𝑦𝑦 𝑥𝑥 − 𝑦𝑦 − 𝑥𝑥

4. Could a circle given by the equation ( 5) + ( 1) = 25 have tangent lines given by the equations 4 3 4 = ( 1) 5 = ( 8) 2 2 3 and 4 𝑥𝑥 − ? Explain𝑦𝑦 −how you know. 𝑦𝑦 − 𝑥𝑥 − 𝑦𝑦 − 𝑥𝑥 −

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M5 GEOMETRY

Lesson Summary

THEOREM: A tangent line to a circle is perpendicular to the radius of the circle drawn to the point of tangency.

Relevant Vocabulary

TANGENT TO A CIRCLE: A tangent line to a circle is a line in the same plane that intersects the circle in one and only one point. This point is called the point of tangency.

Problem Set

1. Consider the circle ( 1) + ( 2) = 16. There are two lines tangent to this circle having a slope of 0. a. Find the coordinates of2 the points2 of tangency. 𝑥𝑥 − 𝑦𝑦 − b. Find the equations of the two tangent lines.

2 4 + + 10 + 13 = 0 2. Consider the circle . There are two lines tangent to this circle having a slope of 3. 2 2 a. Find the coordinates𝑥𝑥 − 𝑥𝑥 of the𝑦𝑦 two points𝑦𝑦 of tangency. b. Find the equations of the two tangent lines.

3. What are the coordinates of the points of tangency of the two tangent lines through the point (1, 1) each tangent to the circle + = 1? 2 2 𝑥𝑥 𝑦𝑦 4. What are the coordinates of the points of tangency of the two tangent lines through the point ( 1, 1) each tangent to the circle + = 1? − − 2 2 𝑥𝑥 𝑦𝑦 5. What is the equation of the tangent line to the circle + = 1 through the point (6, 0)? 2 2 𝑥𝑥 𝑦𝑦 6. D’Andre said that a circle with equation ( 2) + ( 7) = 13 has a tangent line represented by the equation 3 5 = ( + 1) 2 2 2 . Is he correct? Explain.𝑥𝑥 − 𝑦𝑦 − 𝑦𝑦 − − 𝑥𝑥

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M5 GEOMETRY

8 1 = ( + 10) 7. Kamal gives the following proof that 9 is the equation of a line that is tangent to a circle given by 𝑦𝑦 − ( + 1𝑥𝑥) + ( 9) = 145. 2 2 The circle has center ( 1, 9) and radius 12𝑥𝑥.04. The point𝑦𝑦 − ( 10, 1) is on the circle because − ( 10 + 1) + (1 9) = (−9) + ( 8) = 145. 9 1 8 2 2 2 2 8 − = − − − 1 = ( + 10) The slope of the radius is 1 + 10 9; therefore, the equation of the tangent line is 9 . − a. Kerry said that Kamal has made an error. What was Kamal’s error? Explain what he did wrong. − 𝑦𝑦 − 𝑥𝑥 b. What should the equation for the tangent line be?

8. Describe a similarity transformation that maps a circle given by + 6 + 2 = 71 to a circle of radius 3 that is tangent to both axes in the first quadrant. 2 2 𝑥𝑥 𝑥𝑥 𝑦𝑦 − 𝑦𝑦

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY

Lesson 20: Cyclic Quadrilaterals

Classwork Opening Exercise Given cyclic quadrilateral shown in the diagram, prove that + = 180°.

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑥𝑥 𝑦𝑦

Example

Given quadrilateral with + = 180°, prove that quadrilateral is cyclic; in other words, prove that points , , , and lie on the same circle. 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑚𝑚∠𝐴𝐴 𝑚𝑚∠𝐶𝐶 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐴𝐴 𝐵𝐵 𝐶𝐶 𝐷𝐷

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Exercises 1. Assume that vertex lies inside the circle as shown in the diagram. Use a similar argument to Example 1 to show that vertex cannot lie inside the circle. 𝐷𝐷′′ 𝐷𝐷′′

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2. Quadrilateral is a cyclic quadrilateral. Explain why ~ .

𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 △ 𝑃𝑃𝑃𝑃𝑃𝑃 △ 𝑆𝑆𝑆𝑆𝑆𝑆

3. A cyclic quadrilateral has perpendicular diagonals. What is the area of the quadrilateral in terms of , , , and as shown? 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑑𝑑

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4. Show that the triangle in the diagram has area sin( ). 1 2 𝑎𝑎𝑎𝑎 𝑤𝑤

1 (180 )° sin( ) 5. Show that the triangle with obtuse angle has area 2 . − 𝑤𝑤 𝑎𝑎𝑎𝑎 𝑤𝑤

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1 Area = ( + )( + ) sin( ) 6. Show that the area of the cyclic quadrilateral shown in the diagram is 2 . 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑑𝑑 𝑤𝑤

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Lesson Summary

THEOREMS: Given a convex quadrilateral, the quadrilateral is cyclic if and only if one pair of opposite angles is supplementary. The area of a triangle with side lengths and and acute included angle with degree measure : 1 𝑎𝑎 𝑏𝑏Area = sin( ). 𝑤𝑤 2 𝑎𝑎𝑎𝑎 ⋅ 𝑤𝑤 The area of a cyclic quadrilateral whose diagonals and intersect to form an acute or right angle with degree measure : 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴�� 𝐵𝐵𝐵𝐵�� 1 𝑤𝑤 Area( ) = sin( ). 2 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 ⋅ 𝐴𝐴𝐴𝐴 ⋅ 𝐵𝐵𝐵𝐵 ⋅ 𝑤𝑤 Relevant Vocabulary

CYCLIC QUADRILATERAL: A quadrilateral inscribed in a circle is called a cyclic quadrilateral.

Problem Set

1. Quadrilateral is cyclic, is the center of the circle, and = 130°. Find .

𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑂𝑂 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵

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2. Quadrilateral is cyclic, = 8, = 6, = 3, and = 130°. Find the area of quadrilateral .

𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝐴𝐴𝐴𝐴 𝐹𝐹𝐹𝐹 𝑋𝑋𝑋𝑋 𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹

3. In the diagram below, and = 72°. Find the value of and .

�𝐵𝐵��𝐵𝐵� ∥ �𝐶𝐶𝐶𝐶�� 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 𝑠𝑠 𝑡𝑡

4. In the diagram below, is the diameter, = 25°, and . Find and .

�𝐵𝐵𝐵𝐵�� 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 �𝐶𝐶𝐶𝐶�� ≅ �𝐷𝐷𝐷𝐷�� 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 𝑚𝑚∠𝐸𝐸𝐸𝐸𝐸𝐸

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY

5. In circle , = 15°. Find .

𝐴𝐴 𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵

6. Given the diagram below, is the center of the circle. If = 112°, find .

𝑂𝑂 𝑚𝑚∠𝑁𝑁𝑁𝑁𝑁𝑁 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃

7. Given the angle measures as indicated in the diagram below, prove that vertices , , , and lie on a circle.

𝐶𝐶 𝐵𝐵 𝐸𝐸 𝐷𝐷

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8. In the diagram below, quadrilateral is cyclic. Find the value of .

𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽 𝑛𝑛

9. Do all four perpendicular bisectors of the sides of a cyclic quadrilateral pass through a common point? Explain.

10. The circles in the diagram below intersect at points and . If = 100° and = 70°, find and . 𝐴𝐴 𝐵𝐵 𝑚𝑚∠𝐹𝐹𝐹𝐹𝐹𝐹 𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻 𝑚𝑚∠𝐺𝐺𝐺𝐺𝐺𝐺 𝑚𝑚∠𝐸𝐸𝐸𝐸𝐸𝐸

11. A quadrilateral is called bicentric if it is both cyclic and possesses an inscribed circle. (See diagram to the right.) a. What can be concluded about the opposite angles of a bicentric quadrilateral? Explain. b. Each side of the quadrilateral is tangent to the inscribed circle. What does this tell us about the segments contained in the sides of the quadrilateral? c. Based on the relationships highlighted in part (b), there are four pairs of congruent segments in the diagram. Label segments of equal length with , , , and . d. What do you notice about the opposite sides of the bicentric 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑑𝑑 quadrilateral?

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY

12. Quadrilateral is cyclic such that is the diameter of the circle. If , prove that is a right angle, and show that , , , and lie on a circle. 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �𝑃𝑃𝑃𝑃�� ∠𝑄𝑄𝑄𝑄𝑄𝑄 ≅ ∠𝑄𝑄𝑄𝑄𝑄𝑄 ∠𝑃𝑃𝑃𝑃𝑃𝑃 𝑆𝑆 𝑋𝑋 𝑇𝑇 𝑃𝑃

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 21 M5 GEOMETRY

Lesson 21: Ptolemy’s Theorem

Classwork Opening Exercise Ptolemy’s theorem says that for a cyclic quadrilateral , = + .

With ruler and a compass, draw an example of a cyclic𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 quadrilateral.𝐴𝐴𝐴𝐴 ⋅ 𝐵𝐵𝐵𝐵 Label𝐴𝐴𝐴𝐴 its⋅ 𝐶𝐶𝐶𝐶vertices𝐵𝐵𝐵𝐵 ⋅, 𝐴𝐴𝐴𝐴, , and . Draw the two diagonals and . 𝐴𝐴 𝐵𝐵 𝐶𝐶 𝐷𝐷 �𝐴𝐴𝐴𝐴�� 𝐵𝐵𝐵𝐵��

With a ruler, test whether or not the claim that = + seems to hold true.

Repeat for a second example of a cyclic quadrilateral.𝐴𝐴𝐴𝐴 ⋅ 𝐵𝐵𝐵𝐵 𝐴𝐴𝐴𝐴 ⋅ 𝐶𝐶𝐶𝐶 𝐵𝐵𝐵𝐵 ⋅ 𝐴𝐴𝐴𝐴 Challenge: Draw a cyclic quadrilateral with one side of length zero. What shape is this cyclic quadrilateral? Does Ptolemy’s claim hold true for it?

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 21 M5 GEOMETRY

Exploratory Challenge: A Journey to Ptolemy’s Theorem The diagram shows cyclic quadrilateral with diagonals and intersecting to form an acute angle with degree measure . = , = �� , = , and = . 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵 𝑤𝑤 𝐴𝐴𝐴𝐴 𝑎𝑎 𝐵𝐵𝐵𝐵 𝑏𝑏 𝐶𝐶𝐶𝐶a. 𝑐𝑐From 𝐷𝐷𝐷𝐷the last𝑑𝑑 lesson, what is the area of quadrilateral in terms of the lengths of its diagonals and the angle ? Remember this formula for later. 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑤𝑤

b. Explain why one of the angles, or , has a measure less than or equal to 90°.

∠𝐵𝐵𝐵𝐵𝐵𝐵 ∠𝐵𝐵𝐵𝐵𝐵𝐵

c. Let’s assume that in our diagram is the angle with a measure less than or equal to 90°. Call its measure degrees. What is the area of triangle in terms of , , and ? What is the area of triangle in terms of , , and ? What∠𝐵𝐵𝐵𝐵𝐵𝐵 is the area of quadrilateral in terms of , , , , and ? 𝑣𝑣 𝐵𝐵𝐵𝐵𝐵𝐵 𝑏𝑏 𝑐𝑐 𝑣𝑣 𝐵𝐵𝐵𝐵𝐵𝐵 𝑎𝑎 𝑑𝑑 𝑣𝑣 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑑𝑑 𝑣𝑣

d. We now have two different expressions representing the area of the same cyclic quadrilateral . Does it seem to you that we are close to a proof of Ptolemy’s claim? 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 21 M5 GEOMETRY

e. Trace the circle and points , , , and onto a sheet of patty paper. Reflect triangle about the perpendicular bisector of diagonal . Let , , and be the images of the points , , and , respectively. 𝐴𝐴 𝐵𝐵 𝐶𝐶 𝐷𝐷 𝐴𝐴𝐴𝐴𝐴𝐴 i. What does the reflection do with points and ? �𝐴𝐴𝐴𝐴�� 𝐴𝐴′ 𝐵𝐵′ 𝐶𝐶′ 𝐴𝐴 𝐵𝐵 𝐶𝐶 𝐴𝐴 𝐶𝐶

ii. Is it correct to draw as on the circle? Explain why or why not.

𝐵𝐵′

iii. Explain why quadrilateral has the same area as quadrilateral .

𝐴𝐴𝐴𝐴′𝐶𝐶𝐶𝐶 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴

f. The diagram shows angles having degree measures , , , , and . Find and label any other angles having degree measures , , , , or , and justify your answers. 𝑢𝑢 𝑤𝑤 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑢𝑢 𝑤𝑤 𝑥𝑥 𝑦𝑦 𝑧𝑧

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g. Explain why = + in your diagram from part (f).

𝑤𝑤 𝑢𝑢 𝑧𝑧

h. Identify angles of measures , , , , and in your diagram of the cyclic quadrilateral from part (e).

𝑢𝑢 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑤𝑤 𝐴𝐴𝐴𝐴′𝐶𝐶𝐶𝐶

i. Write a formula for the area of triangle in terms of , , and . Write a formula for the area of triangle in terms of , , and . ′ ′ 𝐵𝐵 𝐴𝐴𝐴𝐴 𝑏𝑏 𝑑𝑑 𝑤𝑤 𝐵𝐵 𝐶𝐶𝐶𝐶 𝑎𝑎 𝑐𝑐 𝑤𝑤

j. Based on the results of part (i), write a formula for the area of cyclic quadrilateral in terms of , , , , and . 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑑𝑑 𝑤𝑤

k. Going back to part (a), now establish Ptolemy’s theorem.

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Lesson Summary

THEOREM:

PTOLEMY’S THEOREM: For a cyclic quadrilateral , = + .

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴 ⋅ 𝐵𝐵𝐵𝐵 𝐴𝐴𝐴𝐴 ⋅ 𝐶𝐶𝐶𝐶 𝐵𝐵𝐵𝐵 ⋅ 𝐴𝐴𝐴𝐴 Relevant Vocabulary

CYCLIC QUADRILATERAL: A quadrilateral with all vertices lying on a circle is known as a cyclic quadrilateral.

Problem Set

1. An equilateral triangle is inscribed in a circle. If is a point on the circle, what does Ptolemy’s theorem have to say about the distances from this point to the three vertices of the triangle? 𝑃𝑃

2. Kite is inscribed in a circle. The kite has an area of 108 sq. in., and the ratio of the lengths of the non-congruent adjacent sides is 3 1. What is the perimeter of the kite? 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴

∶

Lesson 21: Ptolemy’s Theorem S.157

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 21 M5 GEOMETRY

3. Draw a right triangle with leg lengths and and hypotenuse length . Draw a rotated copy of the triangle such that the figures form a rectangle. What does Ptolemy have to say about this rectangle? 𝑎𝑎 𝑏𝑏 𝑐𝑐

4. Draw a regular pentagon of side length 1 in a circle. Let be the length of its diagonals. What does Ptolemy’s theorem say about the quadrilateral formed by four of the vertices of the pentagon? 𝑏𝑏

5. The area of the inscribed quadrilateral is 300 mm . Determine the circumference of the circle. 2 √

Lesson 21: Ptolemy’s Theorem S.158

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 21 M5 GEOMETRY

6. Extension: Suppose and are two acute angles, and the circle has a diameter of 1 unit. Find , , , and in terms of and . Apply Ptolemy’s theorem, and determine the exact value of sin(75). 𝑥𝑥 𝑦𝑦 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑑𝑑 a. Explain𝑥𝑥 why𝑦𝑦 equals the diameter of the circle. ( ) 𝑎𝑎 1 b. If the circle hassin a𝑥𝑥 diameter of , what is ? c. Use Thales’ theorem to write the side lengths in the original diagram in 𝑎𝑎 terms of and . d. If one diagonal of the cyclic quadrilateral is 1, what is the other? 𝑥𝑥 𝑦𝑦 e. What does Ptolemy’s theorem give? f. Using the result from part (e), determine the exact value of sin(75).

Lesson 21: Ptolemy’s Theorem S.159