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. 퐴 퐵 c. Mark on the white paper the location of the corner of the colored paper, using a different color than black. 퐴 퐵 Mark that point . See the example below. C 퐶

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 퐷 ( , , …) seem to trace?

퐶 퐷 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. Draw a rotated copy of the triangle underneath it. 퐶 퐷 Label the acute angles in the original퐴 triangle퐵 as and , and label the corresponding angles in the rotated triangle the same. 푥 푦 Todd says ’ is a rectangle. Maryam says ’ is a quadrilateral, but she’s not sure it’s a rectangle. Todd is right but doesn’t know how to explain himself to Maryam. Can you help him out? 퐴퐵퐶퐶 퐴퐵퐶퐶 a. What composite figure is formed by the two triangles? How would you prove it?

i. What is the sum of and ? Why?

푥 푦

Lesson 1: Thales’ Theorem Date: 9/5/14 S.1

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ii. How do we know that the figure whose vertices are the colored points ( , , …) and points and is a rectangle? 퐶 퐷 퐴 퐵

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

퐴 퐵

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 ? 푃 푃 퐶 퐷 푃 퐴 퐵

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 Date: 9/5/14 S.2

Example 1

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

퐴Notice퐵 that,퐶 in the proof in the Exploratory Challenge, you started�퐴퐵�� with a right angle (the corner∠ of퐴퐶퐵 the colored 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.

푃 퐴 퐵 퐶 �퐴퐵�� ∠퐴퐶퐵

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

�푃퐶�� △ 퐴푃퐶 △ 퐵푃퐶

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

d. Label the base angles of as ° and the bases of as °. Express the measure of in terms of ° and °. △ 퐴푃퐶 푏 △ 퐵푃퐶 푎 ∠퐴퐶퐵 푎 푏

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

∠퐴퐶퐵

Lesson 1: Thales’ Theorem Date: 9/5/14 S.3

Exercises 1–2 1. is a diameter of the circle shown. The radius is 12.5 cm, and = 7 cm. a. Find . �퐴퐵�� 퐴퐶 푚∠퐶

b. Find .

퐴퐵

c. Find .

퐵퐶

2. In the circle shown, is a diameter with center . a. Find . �퐵퐶�� 퐴 푚∠퐷퐴퐵

b. Find .

푚∠퐵퐴퐸

c. Find .

푚∠퐷퐴퐸

Lesson 1: Thales’ Theorem Date: 9/5/14 S.4

Lesson Summary

THEOREMS: • THALES’ THEOREM: If , , and are three different points on a circle with a diameter, then is a right angle. 퐴 퐵 퐶 �퐴퐵�� ∠퐴퐶퐵 • 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. ∆퐴퐵퐶 ∠퐶 퐴 퐵 퐶 • Therefore, given distinct points , , and on a circle, is a right triangle with the right angle if �퐴퐵�� and only if is a diameter of the circle. 퐴 퐵 퐶 ∆퐴퐵퐶 ∠퐶 • Given two points and , let point be the midpoint between them. If is a point such that is �퐴퐵�� right, then = = . 퐴 퐵 푃 퐶 ∠퐴퐶퐵 퐵푃 퐴푃 퐶푃 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 . 퐶 푟 퐶 푟 • RADIUS: May refer either to the line segment joining the center of a circle with any point on that circle (a 푟 퐶 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 . The segment is called a chord of . • CENTRAL ANGLE: A central angle of a circle is an angle whose vertex is the center of a circle. 퐶 푃 푄 퐶 �푃푄�� 퐶

Problem Set

1. , , and are three points on a circle, and angle is a right angle. What’s wrong with the picture below? Explain your reasoning. 퐴 퐵 퐶 퐴퐵퐶

Lesson 1: Thales’ Theorem Date: 9/5/14 S.5

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 ?

�퐴퐵�� 퐵퐶 퐴퐶

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

푂 �퐴퐷��

a. Find . b. If = 3 4, what is ? 푚∠퐴푂퐵 푚∠퐴푂퐵 ∶ 푚∠퐶푂퐷 ∶ 푚∠퐵푂퐶

Lesson 1: Thales’ Theorem Date: 9/5/14 S.6

5. is a diameter of a circle, and is another point on the circle. The point lies on the line such that = . Show that = . (Hint: Draw a picture to help you explain your thinking!) �푃푄�� 푀 푅 ⃖푀푄��⃗ 푅푀 푀푄 푚∠푃푅푀 푚∠푃푄푀 6. Inscribe in a circle of diameter 1 such that is a diameter. Explain why: a. sin( ) = . △ 퐴퐵퐶 �퐴퐶�� b. cos( ) = . ∠퐴 퐵퐶 ∠퐴 퐴퐵

Lesson 1: Thales’ Theorem Date: 9/5/14 S.7

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

Classwork Opening Exercise Construct the perpendicular bisector of line segment below (as you did in Module 1).

�퐴퐵��

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

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

Lesson 2: Circle, Chords, Diameters, and Their Relationships Date: 9/5/14 S.8

Exercises 1–6 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: Circle, Chords, Diameters, and Their Relationships Date: 9/5/14 S.9

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, 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.

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: Circle, Chords, Diameters, and Their Relationships Date: 9/5/14 S.10

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: Circle, Chords, Diameters, and Their Relationships Date: 9/5/14 S.11

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 = .

퐴 퐵 퐶 퐴퐵 퐴퐶

Problem Set

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

퐴퐵 푂푀 푂푁 퐶푁

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

�퐴퐶�� ⊥ �퐵퐺�� 퐷퐹�� ⊥ �퐸퐺�� 퐸퐹 퐴퐶

Lesson 2: Circle, Chords, Diameters, and Their Relationships Date: 9/5/14 S.12

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

퐴퐶 퐷퐺 퐸퐺

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

퐴퐵 퐴퐶 퐷퐸

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

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. 퐵 퐷 퐴 퐶 퐴퐶 �퐵퐷�� ⊥ �퐴퐶�� 퐵퐷

Lesson 2: Circle, Chords, Diameters, and Their Relationships Date: 9/5/14 S.13

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. . �퐵퐹�� a. Find . 퐶 퐸 퐶퐸 �퐴퐷�� ⊥ �퐵퐹�� b. Find . 퐴퐷 퐵퐹

8. In the figure, is the center of the circle, and = . Prove that bisects . 퐴 퐶퐵 퐶퐷 �퐴퐶�� ∠퐵퐶퐷

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: Circle, Chords, Diameters, and Their Relationships Date: 9/5/14 S.14

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 . �퐴퐵�� • Mark the point of intersection of the perpendicular chord and the circle �퐴퐵�� 퐵 as point . • is a diameter of the circle. Construct a second diameter in the same 퐶 way. 퐴퐶�� • 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 Date: 9/5/14 S.15

Exercises 1–5 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.

b. List all the symmetries this diagram possesses.

Lesson 3: Rectangles Inscribed in Circles Date: 9/5/14 S.16

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 between the diagonals of the rectangle in the original diagram was 40°.

Lesson 3: Rectangles Inscribed in Circles Date: 9/5/14 S.17

5. Challenge: Show that the 3 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. (This is called the side-splitter theorem.) a. Draw the perpendicular bisectors of and . b. Label the point where they meet . What is point ? �퐴퐵�� 퐴퐶�� 푃 푃

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? 푃

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

퐵 퐵′

e. Using what you have learned about angles, chords, and their relationships, what does the position of point depend on? Why? 푃

Lesson 3: Rectangles Inscribed in Circles Date: 9/5/14 S.18

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

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 .

퐵퐶퐷퐸 퐴 퐷퐸 퐵퐸 퐴퐸

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

Lesson 3: Rectangles Inscribed in Circles Date: 9/5/14 S.19

5. In the figure, and are parallel chords 14 cm apart. = 12 cm, = 10 cm, and . Find . 퐷퐹�� 퐵퐺�� 퐷퐹 퐴퐵 퐸퐻�� ⊥ �퐵퐺�� 퐵퐺

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

Lesson 3: Rectangles Inscribed in Circles Date: 9/5/14 S.20

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 Date: 9/5/14 S.21

Exploratory Challenge 1 Your teacher will provide you with a straight edge, a sheet of colored paper in the shape of a trapezoid, and a sheet of plain white paper. • Draw 2 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 퐴 퐵 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 . 퐴 퐵 • Repeat several times. Name the points , , …. 퐶 퐷 퐸

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? 퐴 퐵 퐶 퐷 퐸

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? 퐴 퐵

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

퐷 ∠퐵퐷퐶 b. is called an inscribed angle. Can you explain why?

∠퐵퐷퐶

Lesson 4: Experiments with Inscribed Angles Date: 9/5/14 S.22

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

퐵퐶�

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 angle , and cut it out. Compare and . What appears to be true about the two angles? 퐸 ∠퐵퐸퐶 ∠퐵퐷퐶 ∠퐵퐸퐶

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 Date: 9/5/14 S.23

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

퐵 퐶

b. Is an inscribed angle? Explain.

∠퐵퐴퐶

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 Example 2?

∠퐵퐴퐶

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

Lesson 4: Experiments with Inscribed Angles Date: 9/5/14 S.24

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 퐶are in the interior of 푂 . The major퐴 arc퐵 is the set containing , , and all퐶 points of that lie in the exterior of . 퐴 퐵 퐶 ∠퐴푂퐵 퐴 퐵 • INSCRIBED ANGLE: 퐶An inscribed angle is an angle∠퐴푂퐵 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. Using a protractor, measure both the inscribed angle and the central angle shown on the circle below.

= ______= ______

푚∠퐵퐶퐷 푚∠퐵퐴퐷

Lesson 4: Experiments with Inscribed Angles Date: 9/5/14 S.25

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

= ______= ______

푚∠퐵퐷퐶 푚∠퐵퐴퐶 3. Using a protractor, measure both the inscribed angle and the central angle shown on the circle below.

= ______= ______

푚∠퐵퐷퐶 푚∠퐵퐴퐶 4. What relationship between the measure of the inscribed angle and the measure of the central angle that intercept the same arc is illustrated by these examples?

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

Lesson 4: Experiments with Inscribed Angles Date: 9/5/14 S.26

6. Prove that = 2 in the diagram below.

푦 푥

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 “danger푅 angle.” What퐵 happens to as the ship gets closer to shore and moves away from shore? Why do you think a larger angle is dangerous?푆 ∠푅푆퐵 휃 휃

Red ( )

𝑹𝑹

Blue ( )

𝑩𝑩

Lesson 4: Experiments with Inscribed Angles Date: 9/5/14 S.27

Lesson 5: Inscribed Angle Theorem and its Applications

Classwork Opening Exercise 1. and are points on a circle with center . a. Draw a point on the circle so that is a diameter. Then draw 퐴 퐶 푂 the angle . 퐵 �퐴퐵�� ∠퐴퐵퐶 b. What angle in your diagram is an inscribed angle?

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

d. What is the intercepted arc of angle ?

∠퐴퐵퐶

e. What is the intercepted arc of ?

∠퐴푂퐶

2. The measure of the inscribed angle is and the measure of the central angle is . Find in terms of . 푥 푦 푚∠퐶퐴퐵 푥

Lesson 5: Inscribe Angle Theorem and its Applications Date: 9/5/14 S.28

Example 1

and are points on a circle with center .

퐴 퐶 푂

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

∠퐶푂퐴

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

퐴푂퐶

c. What can you conclude about and ? Why?

푚∠푂퐶퐴 푚∠푂퐴퐶

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

퐵 푂 ∠퐴퐵퐶 e. What is the intercepted arc of angle ? Color it green.

∠퐴퐵퐶

f. What do you notice about arc ?

퐴퐶�

Lesson 5: Inscribe Angle Theorem and its Applications Date: 9/5/14 S.29

g. Let the measure of be and the measure of be . Can you prove that = 2 ? (Hint: Draw the diameter that contains point .) ∠퐴퐵퐶 푥 ∠퐴푂퐶 푦 푦 푥 퐵

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 .

퐴 퐶 푂

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

퐵 푂 ∠퐴퐵퐶 b. What is the intercepted arc of angle ? Color it yellow.

∠퐴퐵퐶

c. Let the measure of be , and the measure of be . Can you prove that = 2 ? (Hint: Draw the diameter that contains point .) ∠퐴퐵퐶 푥 ∠퐴푂퐶 푦 푦 푥 퐵

Lesson 5: Inscribe Angle Theorem and its Applications Date: 9/5/14 S.30

d. Does your conclusion support the inscribed angle theorem?

e. Have we finished proving the inscribed angle theorem?

Exercises 1–5 1. Find the measure of the angle with measure .

a. = 25° b. 푥 = 15° c. = 90° 푚∠퐷 푚∠퐷 푚∠퐵퐴퐶

x x x

Lesson 5: Inscribe Angle Theorem and its Applications Date: 9/5/14 S.31

d. = 32° e. f. = 19°

푚∠퐵 푚∠퐷

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

Lesson 5: Inscribe Angle Theorem and its Applications Date: 9/5/14 S.32

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. 푥 푦 푦 푥

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

푥

Lesson 5: Inscribe Angle Theorem and its Applications Date: 9/5/14 S.33

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

c. d.

e. f.

Lesson 5: Inscribe Angle Theorem and its Applications Date: 9/5/14 S.34

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.

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 a right angle is the semicircle in the interior of the angle.

Problem Set

Find the value of in each exercise.

1. 푥 2.

Lesson 5: Inscribe Angle Theorem and its Applications Date: 9/5/14 S.35

3. 4.

5. 6.

Lesson 5: Inscribe Angle Theorem and its Applications Date: 9/5/14 S.36

7. 8.

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 . 퐸 퐹 퐷 �퐹퐺�� 퐻 푥 푦

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

Lesson 5: Inscribe Angle Theorem and its Applications Date: 9/5/14 S.37

10. In the figure below, and intersect at point E. �� Prove: + 퐸퐷 퐵퐶= 2( )

푚∠퐷퐴퐵 푚∠퐸퐴퐶 푚∠퐵퐹퐷

PROOF: Join .

퐵퐸�� 1 = ( ______) 2 푚∠퐵퐸퐷 푚∠ 1 = ( ______) 2 푚∠퐸퐵퐶 푚∠

In ,

∆퐸퐵퐹 + = ______

푚∠퐵퐸퐹 푚∠퐸퐵퐹 푚∠

1 1 ( ______) + ( ______) = ______2 2 푚∠ 푚∠ 푚∠

+ = 2( )

∴ 푚∠퐷퐴퐵 푚∠퐸퐴퐶 푚∠퐵퐹퐷

Lesson 5: Inscribe Angle Theorem and its Applications Date: 9/5/14 S.38

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 . �퐷퐸�� 퐴퐵�� 퐶 푚∠퐷퐴퐸 ⁰ 푚∠퐷퐶퐴 ⁰ 푚∠퐷퐸퐴

Let = , =

푚∠퐷퐸퐴 푦 푚∠퐸퐴퐸 푥 In , = Reason

∆퐴퐵퐷 푚∠퐷퐵퐴 = Reason

푚∠퐴퐷퐵 46 + + + 90 = Reason

∴ 푥 푦 + =

푥 푦 In , = + 32 Reason

+∆퐴퐶퐸+ 32푦= 푥 Reason 푥 푥 =

푥 = 푦 = 푚∠퐷퐸퐴

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Date: 9/5/14 S.39

Exercises 1–4 Find the value in each figure below, and describe how you arrived at the answer.

1. Hint: Thales’푥 theorem 2.

3. 4.

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Date: 9/5/14 S.40

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 line , and angles and are congruent, then , , ’, and all lie on the same circle. 퐴 퐵 퐵 퐶 퐵 퐵 ⃖퐴퐶��⃗ ∠퐴퐵퐶 ∠퐴퐵′퐶 퐴 퐵 퐵 퐶 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.

Problem Set

In Problems 1–5, find the value . 1. 푥

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Date: 9/5/14 S.41

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Date: 9/5/14 S.42

6. If = , express in terms of .

퐵퐹 퐹퐶 푦 푥

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Date: 9/5/14 S.43

7. a. Find the value .

푥

b. Suppose the = . Prove that = 3 .

푚∠퐶 푎⁰ 푚∠퐷퐸퐵 푎⁰

8. In the figure below, three identical circles meet at , and , E respectively. = . , , and , , lie on straight lines. 퐵 퐹 퐶 퐵퐹 퐶퐸 퐴 퐵 퐶 퐹 퐸 퐷 Prove is a parallelogram.

퐴퐶퐷퐹

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Date: 9/5/14 S.44

PROOF:

Join and .

퐵퐸 퐶퐹 = Reason: ______

퐵퐹 퐶퐸 = ______= ______= ______= Reason: ______

푎 푑 ______= ______

Alternate angles are equal.

퐴퐶��‖퐹퐷�� ______= ______

Corresponding angles are equal.

퐴퐹��‖퐵퐸�� ______= ______

Corresponding angles are equal.

퐵퐸��‖�퐶퐷��

퐴퐹��‖퐵퐸��‖�퐶퐷�� is a parallelogram.

퐴퐶퐷퐹

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Date: 9/5/14 S.45

Lesson 7: The Angle Measure of an Arc

Classwork Opening Exercise If the measure of is 17°, name 3 other angles that have the same measure and explain why. ∠퐺퐵퐹

What is the measure of ? Explain.

∠퐺퐴퐹

Can you find the measure of ? Explain.

∠퐵퐴퐷

Lesson 7: The Angle Measure of an Arc Date: 9/5/14 S.46

Example 1

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

퐴 퐶

Exercises 1–4

1. In circle , : : : = 1: 2: 3: 4. Find a. 퐴 퐵퐶� 퐶퐸� 퐸퐷� �퐷퐵� 푚∠퐵퐴퐶

푚∠퐷퐴퐸

푚퐷�퐵

푚퐶�퐸퐷

Lesson 7: The Angle Measure of an Arc Date: 9/5/14 S.47

2. In circle , = . Find a. 퐵 퐴퐵 퐶퐷 푚퐶퐷�

푚퐶퐴�퐷

푚퐴퐷�

3. In circle , is a diameter and = 100 . If = 2 , find a. 퐴 �퐵퐶�� 푚∠퐷퐴퐶 ⁰ 푚퐸퐶� 푚퐵퐷� 푚∠퐵퐴퐸

푚퐸�퐶

푚퐷�퐸퐶

4. Given circle A with = 37 , find a. 푚∠퐶퐴퐷 ⁰ 푚퐶퐵퐷�

푚∠퐶퐵퐷

푚∠퐶퐸퐷

Lesson 7: The Angle Measure of an Arc Date: 9/5/14 S.48

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 are퐶 in the interior of .푂 The major퐴 arc is퐵 the set containing , , and퐶 all points of that lie in the exterior of . 퐴 퐵 퐶 ∠퐴푂퐵 퐴 퐵 • SEMICIRCLE: In a circle,퐶 let and be the endpoints∠퐴푂퐵 of a diameter. A semicircle is the set containing , , and all points of the circle that lie in a given half-plane of the line determined by the diameter. 퐴 퐵 퐴 퐵 • 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.

Problem Set

1. Given circle with = 50 , a. Name a central angle. 퐴 푚∠퐶퐴퐷 ⁰ b. Name an inscribed angle. c. Name a chord. d. Name a minor arc. e. Name a major arc. f. Find , g. Find . 푚퐶퐷� h. Find . 푚퐶퐵퐷� 푚∠퐶퐵퐷

Lesson 7: The Angle Measure of an Arc Date: 9/5/14 S.49

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

퐴

3. Given circle , find a. 퐴 b. 푚∠퐵퐴퐷 c. 푚∠퐶퐴퐵 d. 푚퐵퐶� e. 푚퐵퐷� 푚퐵퐶퐷�

4. Find the angle measure of angle .

푥

Lesson 7: The Angle Measure of an Arc Date: 9/5/14 S.50

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

푚∠퐵퐴퐶 ⁰ 푚∠퐵퐸퐷 ⁰ 푚∠퐷퐸퐶

6. In the figure = 74 , and = 42 . is the midpoint of and is the midpoint of . Find and 0 . 푚∠퐵퐶퐷 ⁰ 푚∠퐵퐷퐶 퐾 퐶퐵� 퐽Solution: Join , 퐵퐷�, , 푚∠, 퐾퐵퐷, and . 푚∠퐶퐾퐽 퐵퐾 퐾퐶 퐾퐷 퐾퐽 퐽퐶 퐽퐷 = ______

푚�퐵퐾 푚퐾퐶� = = 21 ______42 푚∠퐾퐷퐶 2 ⁰ a = ______

In , b = ______

∆퐵퐶퐷 c = ______

= ______

푚�퐵퐽 푚퐽퐷� = ______

푚∠퐽퐶퐷 d = ______

= + = ______

푚∠퐾퐵퐷 푎 푏 = + = ______

푚∠퐶퐾퐽 푐 푑

Lesson 7: The Angle Measure of an Arc Date: 9/5/14 S.51

Lesson 8: Arcs and Chords

Classwork Opening Exercise Given circle with , = 6, and = 10. Find and . Explain your work. 퐴 �퐵퐶�� ⊥ �퐷퐸�� 퐹퐴 퐴퐶 퐵퐹 퐷퐸

Exercises 1. Given circle with = 54 and , find . Explain your work. 퐴 푚퐵퐶� ⁰ ∠퐶퐷퐵 ≅ ∠퐷퐵퐸 푚퐷퐸�

Lesson 8: Arcs and Chords Date: 9/5/14 S.52

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 .

퐶퐷� 퐸퐷�

4. = and : : = 1: 2: 4. Find a. 푚퐶퐵� 푚퐸퐷� 푚퐵퐶� 푚퐵퐷� 푚퐸퐶� 푚∠퐵퐶퐹

푚∠퐸퐷퐹

푚∠퐶퐹퐸

Lesson 8: Arcs and Chords Date: 9/5/14 S.53

5. is a diameter of circle . : : = 1: 3: 5. Find a. 퐵퐶�� 퐴 푚퐵퐷� 푚퐷퐸� 푚퐸퐶� 푚퐵퐷�

푚퐷�퐸퐶

푚퐸�퐶퐵

Lesson 8: Arcs and Chords Date: 9/5/14 S.54

Lesson Summary

THEOREMS: • Congruent chords have congruent arcs. • Congruent arcs have congruent chords. • Arcs between parallel chords are congruent.

Problem Set

1. Find a. b. 푚퐶�퐸 c. 푚퐵퐷� 푚퐸퐷�

2. In circle , is a diameter, = , and = 32 . 퐴 �퐵퐶�� 푚퐶퐸� 푚퐸퐷� 푚∠퐶퐴퐸 a. Find . ⁰ b. Find . 푚∠퐶퐴퐷 푚∠퐴퐷퐶

Lesson 8: Arcs and Chords Date: 9/5/14 S.55

3. In circle , is a diameter, 2 = , and . Find . 퐴 �퐵퐶�� 푚퐶퐸� 푚퐸퐷� �퐵퐶��‖�퐷퐸�� 푚∠퐶퐷퐸

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

a. Find퐴 �퐵퐶�� . 퐶퐸� ⁰ b. Find . 푚퐶퐷� c. Find 푚∠퐷퐵퐸 푚∠퐷퐶퐸

5. In the circle given, . Prove .

퐵퐶� ≅ 퐸퐷� �퐵퐸�� ≅ �퐷퐶��

Lesson 8: Arcs and Chords Date: 9/5/14 S.56

6. Given circle with , show .

퐴 �퐴퐷��‖퐶퐸�� 퐵퐷� ≅ 퐷퐸�

7. In circle , is a radius and and = 54 . Find . Complete the proof. = ______퐴 �퐴퐵�� 퐵퐶� ≅ 퐵퐷� 푚∠퐶퐴퐷 ⁰ 푚∠퐴퐵퐶 퐵퐶 퐵퐷 _____ = ______

푚∠ 푚∠ + + = ______

푚∠퐵퐴퐶 푚∠퐶퐴퐷 푚∠퐵퐴퐷 2 _____ + 54 = 360 ______

푚∠ ⁰ ⁰ = ______

푚∠퐵퐴퐶 = ______

퐴퐵 퐴퐶 _____ = ______

푚∠ 푚∠ 2 + = ______

푚∠퐴퐵퐶 푚∠퐵퐴퐶 = _____

푚∠퐴퐵퐶

Lesson 8: Arcs and Chords Date: 9/5/14 S.57

Lesson 9: Arc Length and Areas of Sectors

Classwork Example 1

a. What is the length of the arc of degree measure in a circle of radius 10 cm?

60˚

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. 퐴 푚∠퐴 ⁰ ∠퐴

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

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

푥˚ 푟

Lesson 9: Arc Length and Areas of Sectors Date: 9/5/14 S.58

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

Exercise 1 1. The radius of the following circle is 36 cm, and the = . a. What is the arc length of ? 푚∠퐴퐵퐶 60˚ 퐴퐶�

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

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. 퐴퐵� 푂 푟 �푂푃�� 푃 퐴퐵�

Lesson 9: Arc Length and Areas of Sectors Date: 9/5/14 S.59

Example 2

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

푂

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 degrees.

e. Circle has a minor arc with an angle measure of . Sector has an area of 24 . What is the radius of circle ? 푂 퐴퐵� 60˚ 퐴푂퐵 휋 푂

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

푥˚ 푟

Lesson 9: Arc Length and Areas of Sectors Date: 9/5/14 S.60

Exercises 2–3 2. The area of sector in the following image is 28 . Find the measurement of the central angle labeled . 퐴푂퐵 휋 푥˚

3. In the following figure, circle has a radius of 8 cm, = and = = 10 cm. Find: 푂 푚∠퐴푂퐶 108˚ a. 퐴�퐵 퐴퐶� ∠푂퐴퐵

퐵퐶�

c. Area of sector

퐵푂퐶

Lesson 9: Arc Length and Areas of Sectors Date: 9/5/14 S.61

Lesson Summary

Relevant Vocabulary

• ARC: An arc is any of the following three figures—a minor arc, a major arc, or a semicircle.

1 • 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 .푂 The major퐴 arc퐵 is the set containing , , and all points of the circle that lie in the exterior of . 퐴 퐵 퐴 퐵 ∠퐴푂퐵 퐴 퐵 • RADIAN: A radian is the measure∠ 퐴푂퐵of the central angle of a sector of a circle with arc length of one radius length.

• SECTOR: Let arc be an arc of a circle with center and radius . The union of the segments , where is any point on the arc , is called a sector. The arc is called the arc of the sector, and is called its � radius. 퐴퐵 푂 푟 푂푃 푃 퐴퐵� 퐴퐵� 푟 • SEMICIRCLE: In a circle, let and be the endpoints of a diameter. A semicircle is the set containing , , and all points of the circle that lie in a given half-plane of the line determined by the diameter. 퐴 퐵 퐴 퐵

Problem Set

1. and are points on the circle of radius 5 cm and the measure of arc is . Find, to one decimal place each of the following: 푃 푄 푃푄� 72˚ a. The length of arc b. Find the ratio of the arc length to the radius of the circle. 푃푄�

c. The length of chord

푃푄

Lesson 9: Arc Length and Areas of Sectors Date: 9/5/14 S.62

d. The distance of the chord from the center of the circle.

푃푄

e. The perimeter of sector . f. The area of the wedge between푃푂푄 the chord and the arc g. The perimeter of this wedge. 푃푄 푃푄�

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

45˚ 휋 3. Arcs and CD both have an angle measure of , but their arc lengths are not the same. = 4 and = 2. 퐴퐵� � 30˚ a. What are the �푂퐵arc�� lengths of퐵퐷�� arcs� and ? b. What is the ratio of the arc length to the radius for all of these arcs? 퐴퐵� 퐶퐷� Explain. c. What are the areas of the sectors and ?

퐴푂퐵 퐶푂퐷 4. In the circles shown, find the value of .

The circles shown have central angles 푥that are equal in measure. a. b.

Lesson 9: Arc Length and Areas of Sectors Date: 9/5/14 S.63

c. d.

5. The concentric circles all have center . The measure of the central angle is 45°. The arc lengths are given. 퐴 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 = 10 cm, find the length of arc ?

푃푄� 푄푅�

Lesson 9: Arc Length and Areas of Sectors Date: 9/5/14 S.64

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 Date: 9/5/14 S.65

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 . Show how to calculate the area of the shaded region and explain the parts of the expression. 푟 푠

Exercises 1–13 1. Find the area of the following annulus.

Lesson 10: Unknown Length and Area Problems Date: 9/5/14 S.66

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 휋

4. An ice cream shop wants to design a super straw to serve with their extra thick milkshakes that is double the width and thickness of a standard straw. A standard straw is 8 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 23 mm long, what is the maximum volume of milkshake that can be in the straw at one time (round to the nearest hundredth)?

c. A large milkshake is 32 ounces (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)?

Lesson 10: Unknown Length and Area Problems Date: 9/5/14 S.67

5. In the circle given, is the diameter and is perpendicular to chord . = 8 cm and = 2 cm. Find , , , the arc length of CEB, and the area of sector CEB (round to the nearest hundredth, if necessary). �퐸퐷�� 퐶퐵�� 퐷퐹 퐹퐸 퐴퐶 퐵퐶 푚∠퐶퐴퐵 � �

6. Given circle with , find the following (round to the nearest hundredth, if necessary): 퐴 ∠퐵퐴퐶 ≅ ∠퐵퐴퐷 a.

푚퐶퐷�

푚퐶�퐵퐷

푚퐵�퐶퐷

d. Arc length

퐶퐷�

e. Arc length

퐶퐵퐷�

f. Arc length

퐵퐶퐷�

g. Area of sector

퐶퐷�

Lesson 10: Unknown Length and Area Problems Date: 9/5/14 S.68

h. Area of sector

퐶퐵퐷�

i. Area of sector

퐵퐶퐷�

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

b. Radius of circle

퐴

c. Area of sector

퐶퐷�

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

푚∠퐶퐴퐷

b. Area of sector

퐶퐷�

Lesson 10: Unknown Length and Area Problems Date: 9/5/14 S.69

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)? Ө Ө

b. What is the value of in degrees?

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 ∆circles퐴퐵퐶 creating the arcs shown. Find the following (round퐷 to퐸 the nearest퐹 hundredth): a. The area of the sector with center .

퐴

Lesson 10: Unknown Length and Area Problems Date: 9/5/14 S.70

b. The area of triangle .

퐴퐵퐶

c. The area of the shaded region.

d. The perimeter of the shaded region.

12. In the figure shown, = = 5 cm, = 2 cm, and = 30°. Find the area in the rectangle, but outside퐴퐶 of퐵퐹 the circles (round to the퐺퐻 nearest hundredth).푚∠HBI

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 red section, the white section, and the blue section (round to the nearest hundredth).

Lesson 10: Unknown Length and Area Problems Date: 9/5/14 S.71

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 � � �� 퐷퐹hundredth).퐵퐺 퐵퐷 퐹퐺

Lesson 10: Unknown Length and Area Problems Date: 9/5/14 S.72

4. Find the radius of the circle, , , and (round to the nearest hundredth). 푥 푦 푧

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 � �� 푚.퐷퐴퐸 퐷퐸 퐶 휋

Lesson 10: Unknown Length and Area Problems Date: 9/5/14 S.73

Lesson 11: Properties of Tangents

Classwork Exercises 1–3 1. and are tangent to circle at points and respectively. Use a two-column proof to prove �� 퐶퐷= . 퐶퐸 퐴 퐷 퐸

푎 푏

2. In circle , the radius is 9 mm and = 12 mm. a. Find . 퐴 퐵퐶 퐴퐶

b. Find the area of .

∆퐴퐶퐷

c. Find the perimeter of quadrilateral .

퐴퐵퐶퐷

Lesson 11: Properties of Tangents Date: 9/5/14 S.74

3. In circle , = 12 and = 13. : = 1: 3. Find a. The radius of the circle. 퐴 퐸퐹 퐴퐸 퐴퐸 퐴퐶

b. (round to the nearest whole number)

퐵퐶

퐸퐶

Lesson 11: Properties of Tangents Date: 9/5/14 S.75

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 . 푂 푟 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 푂 푟 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 . 푂 푟 A point exterior to a circle is said to be outside the circle. 푂 푟

• 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. 퐴퐵 퐵퐶 퐴퐶 퐵퐶⃖��⃗ 퐴 퐵

Lesson 11: Properties of Tangents Date: 9/5/14 S.76

2. is tangent to circle at point . = 9 and = 15. a. Find the radius of the circle. 퐵퐶⃖��⃗ 퐴 퐵 퐷퐶 퐵퐶 b. Find .

퐴퐶

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, 퐶퐷is there퐹퐸 more wood or metal fencing used?

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

푥 퐵

Lesson 11: Properties of Tangents Date: 9/5/14 S.77

5. Line is tangent to the circle at point , and = . Find CD a. ⃖푃퐶��⃗( ) 퐶 퐶퐷 퐷퐸 b. ( ) 푥 푚 � c. ( ) 푦 푚∠퐶퐹퐸 푧 푚∠PCF

6. Construct two lines tangent to circle through point .

퐴 퐵

Lesson 11: Properties of Tangents Date: 9/5/14 S.78

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

푥

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 , (round to the nearest hundredth). 퐸퐹�� 퐴 퐶 퐴 퐶 �퐸퐹�� 푥

Lesson 11: Properties of Tangents Date: 9/5/14 S.79

Lesson 12: Tangent Segments

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

푧

Example 1

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

a. 퐷 ∠퐵퐴퐶

Lesson 12: Tangent Segments Date: 9/5/14 S.80

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

Lesson 12: Tangent Segments Date: 9/5/14 S.81

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

Exercises 1–5 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:

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

Lesson 12: Tangent Segments Date: 9/5/14 S.82

2. An angle is shown below. a. Draw at least 3 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? 푃 푃

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.

Lesson 12: Tangent Segments Date: 9/5/14 S.83

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. 푃 푃

5. Using a straightedge, draw a large triangle .

퐴퐵퐶

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?

Lesson 12: Tangent Segments Date: 9/5/14 S.84

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? 퐴 퐶 b. Draw two tangents from to the circle, and label the tangency points and . Fold your paper along the line 퐶 . What do you notice about the lengths of and ? About the measures of the angles and ? 퐶 퐷 퐸 퐴퐶 퐶퐷 퐶퐸 ∠퐷퐶퐴 c. is the ______of . ∠퐸퐶퐴 d. and are tangent to circle . Find . 퐴퐶 ∠퐷퐶퐸 �퐶퐷�� 퐶퐸�� 퐴 퐴퐶 2. In the figure at right, the three segments are tangent to the circle at points , , and . If = x, find , , and . 2 퐵 퐹 퐺 푦 3 푥 푦 푧

Lesson 12: Tangent Segments Date: 9/5/14 S.85

3. In the figure given, the three segments are tangent to the circle at points , , and . a. Prove = + 퐽 퐼 퐻 b. Find the perimeter of . 퐺퐹 퐺퐽 퐻퐹 ∆퐺퐶퐹

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

퐹 퐵 퐺 퐷퐸

5. is tangent to circle . If points and are the intersection points of circle and any line 퐸퐹⃖��⃗ 퐴 퐶 퐷 parallel to , answer the following. 퐴 a. Does = for any line parallel to ⃖퐸퐹��⃗ ? Explain. 퐶퐺 퐺퐷 b. 퐸퐹Suppose⃖��⃗ that coincides with . Would , , and all coincide with ? ⃖퐶퐷��⃗ ⃖퐸퐹��⃗ c. Suppose , , and have now reached 퐶 퐺 퐷 퐵 , so is tangent to the circle. What is 퐶 퐺 퐷 the angle between the line and ? 퐵 ⃖퐶퐷��⃗ ⃖퐶퐷��⃗ �퐴퐵��

Lesson 12: Tangent Segments Date: 9/5/14 S.86

d. Draw another line tangent to the circle from some point, , in the exterior of the circle. If the point of tangency is point , what is the measure of ? 푃 푇 ∠푃푇퐴 6. The segments are tangent to circle at points and . is a diameter of the circle. 퐴 퐵 퐷 a. Prove . �퐸��퐷� b. Prove quadrilateral is a kite. 퐵퐸��‖퐶퐴�� 퐴퐵퐶퐷

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 퐵퐻by that chord and its inscribed angle퐵 ? ∠퐷퐵퐻 ∠퐷퐶퐵

Lesson 12: Tangent Segments Date: 9/5/14 S.87

Lesson 13: The Inscribed Angle Alternate a Tangent Angle

Classwork Opening Exercise 1. In circle , = 56°, and is a diameter. Find the listed measure, and explain your answer. 퐴 푚퐵퐷� �퐵퐶�� a.

푚∠퐵퐷퐶

푚∠퐵퐶퐷

푚∠퐷퐵퐶

푚∠퐵퐹퐺

푚퐵�퐶

푚퐷�퐶

g. Is the = 56°? Explain.

푚∠퐵퐺퐷

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

∠퐵퐺퐷

Lesson 13: The Inscribed Angle Alternate a Tangent Angle Date: 9/5/14 S.88

Example 1

Diagram 1 Diagram 2

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

푎 푏

Test your conjecture by using a protractor to measure and .

푎 푏

Diagram 1 푎 푏

Diagram 2

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

푎 푏

Lesson 13: The Inscribed Angle Alternate a Tangent Angle Date: 9/5/14 S.89

Now test your conjecture further using the circle below.

푎 푏

Lesson 13: The Inscribed Angle Alternate a Tangent Angle Date: 9/5/14 S.90

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 , 퐴 �퐴퐵��⃗ 퐶 = then . ⃖��⃗ 1 퐴퐶 푎 푚∠퐵퐴퐶 푏 ∠퐵퐴퐶 푎 2 푏

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. 퐴 퐵퐺 a. Draw triangle . What is the measure of ? Explain.

퐴퐵퐶 ∠퐵퐴퐶

b. What is the measure of ? Explain.

∠퐴퐵퐺

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

퐴퐵퐶 푎

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

∠퐵퐴퐶 푎

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

Lesson 13: The Inscribed Angle Alternate a Tangent Angle Date: 9/5/14 S.91

Exercises Find , , , , and/or .

1. 푥 푦 푎 푏 푐

Lesson 13: The Inscribed Angle Alternate a Tangent Angle Date: 9/5/14 S.92

Lesson 13: The Inscribed Angle Alternate a Tangent Angle Date: 9/5/14 S.93

Lesson 13: The Inscribed Angle Alternate a Tangent Angle Date: 9/5/14 S.94

Lesson Summary

THEOREMS:

• CONJECTURE: Let A be a point on a circle, let AB be a tangent ray to the circle, and let C be a point on the AC a = b circle such that is a secant to the circle. �If�� ⃗ and is the angle measure of the arc a = b intercepted by ⃖��⃗ , then . 1 m∠BAC • THE TANGENT-SECANT∠BAC THEOREM: Let2 A be a point on a circle, let AB be a tangent ray to the circle, and let C be a point on the circle such that AC is a secant to the circle. If a = and b is ��⃗ a = b the angle measure of the arc intercepted by⃖� ��⃗ , then . 1 m∠BAC • Suppose AB is a chord of circle C, and AD is a∠BAC tangent segment2 to the circle at point A. If E is any point other than A or B in the arc of C on the opposite side of AB from D, then = . �� m∠BEA m∠BAD Problem Set

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

1. 푎 푏 푐 2. 3.

Lesson 13: The Inscribed Angle Alternate a Tangent Angle Date: 9/5/14 S.95

4. 5. 6.

7. 8. 9.

10. is tangent to circle . is a diameter. Find a. ⃖퐵퐻��⃗ 퐴 �퐷퐹�� b. 푚∠퐵퐶퐷 c. 푚∠퐵퐴퐹 d. 푚∠퐵퐷퐴 e. 푚∠퐹퐵퐻 푚∠퐵퐺퐹

Lesson 13: The Inscribed Angle Alternate a Tangent Angle Date: 9/5/14 S.96

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

퐵퐺�� 퐴 �퐵퐸�� 푓 푎 푑 푐

Lesson 13: The Inscribed Angle Alternate a Tangent Angle Date: 9/5/14 S.97

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 . ⃖퐷퐵��⃗ 푎 푏

b. Is a diameter of the circle? Explain.

퐶퐵��

Exercises 1–2 1. In circle , is a radius, and = 14°. Find , and explain how you know. 푃 �푃푂�� 푚푀푂� 푚∠푀푂푃

Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle Date: 9/5/14 S.98

2. In the circle shown, CE = 55°. Find and EG. Explain your answer. 푚� 푚∠DEF 푚 �

Example 1

a. Find . Justify your answer.

푥

b. Find .

푥

Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle Date: 9/5/14 S.99

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 .

푥 푦 3. 4.

Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle Date: 9/5/14 S.100

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

�퐵퐶�� 푥 푦

7. In the circle shown, is a diameter. : = 2: 1. Prove = 180 using�� a� two-column proof. 3 퐵퐶 퐷퐶 퐵퐸 푦 − 2 푥

Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle Date: 9/5/14 S.101

Lesson Summary

THEOREMS:

. 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

. 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.

Problem Set

In Problems 1–4, find .

푥 1.

Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle Date: 9/5/14 S.102

3. 4.

5. Find ( ) and ( ).

푥 푚퐶퐸� 푦 푚퐷퐺�

Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle Date: 9/5/14 S.103

6. Find the ratio of : .

푚퐸퐹퐶� 푚퐷퐺퐵�

7. is a diameter of circle . Find .

�퐵퐶�� 퐴 푥

Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle Date: 9/5/14 S.104

8. Show that the general formula we discovered in Example 1 also works for central angles. (Hint: Extend the radii to form 2 diameters, and use relationships between central angles and arc measure.)

Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle Date: 9/5/14 S.105

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.

푎 푏 푐

CONJECTURE about the relationship between , , and :

푎 푏 푐

Lesson 15: Secant Angle Theorem, Exterior Case Date: 9/5/14 S.106

2. We will prove the following.

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. We can interpret this statement in terms of the diagram below. Let and be the angle measures of the arcs intercepted by the angles and . Then measure is the average of and ; that is, = . 푏 푐 푏+푐 ∠푆퐴푄 ∠푃퐴푅 푎 푏 푐 푎 2

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.

푏 푐

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 Date: 9/5/14 S.107

Example 1

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

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

푎 푏 푐

Conjecture about the relationship between , , and :

푎 푏 푐

Test your conjecture with another diagram.

Lesson 15: Secant Angle Theorem, Exterior Case Date: 9/5/14 S.108

Exercises Find , , and/or .

1. 푥 푦 푧 2.

3. 4.

Lesson 15: Secant Angle Theorem, Exterior Case Date: 9/5/14 S.109

Lesson Summary: 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’ll summarize the different cases.

The Inscribed Angle Theorem and its Family of Theorems

Relationship between , , , Diagram How the two shapes overlap and 푎 푏 푐 푑

(Inscribed Angle Theorem)

(Secant – Tangent)

Lesson 15: Secant Angle Theorem, Exterior Case Date: 9/5/14 S.110

(Secant Angle Theorem: Interior)

(Secant Angle Theorem: Exterior)

(Two Tangent Lines)

Lesson 15: Secant Angle Theorem, Exterior Case Date: 9/5/14 S.111

Lesson Summary

THEOREMS:

• 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.

• 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 .

풙 풎∠푫푭푬 풎∠푫푮푩

Lesson 15: Secant Angle Theorem, Exterior Case Date: 9/5/14 S.112

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

풎∠푬푪푫 풎∠푫푩푬 풎∠푫푬푩 풎∠푭푮푬 풎∠푭푯푬

5. Find and . 6. The radius of circle is . and are tangent to the circle with = . Find and the area of quadrilateral DAEC �� 풙 풚 rounded to the nearest푨 ퟒhundredth.푫푪 푪푬 푫푪 ퟏퟐ 풎푬푩푫�

7. Find , , and . 8. Find and .

풎푩푮� 풎푮푭� 풎푭푩� 풙 풚

Lesson 15: Secant Angle Theorem, Exterior Case Date: 9/5/14 S.113

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 the segments between the point of intersection of the tangent lines and the circle? Round to the nearest hundredth.

10. and are tangent to circle . Prove = .

�퐷퐶�� 퐸퐶�� 퐴 퐵퐷 퐵퐸

Lesson 15: Secant Angle Theorem, Exterior Case Date: 9/5/14 S.114

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 .

1. 2. 푥

3. 4.

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams Date: 9/5/14 S.115

Example 1

Measure the lengths of the chords in centimeters and record them in the table.

a. s b.

c. d.

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

a 푎 푏 푐 푑

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams Date: 9/5/14 S.116

Example 2

Measure the lengths of the chords in centimeters and record them in the table.

a. b.

c. d.

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

푎 푏 푐 푑 a

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams Date: 9/5/14 S.117

The inscribed angle theorem and its family:

Relationship between Diagram How the two shapes overlap , , and

푎 푏 푐 푑

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams Date: 9/5/14 S.118

Lesson Summary

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

• When secant lines intersect outside of a circle, 푎use ∙ 푏 ( 푐+ ∙ 푑) = ( + ). 푎 푎 푏 푐 푐 푑 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 .

푥 푥

3. < , 1, < . Prove = 3 4. = 6, = 9, = 18. Show = 3.

퐷퐹 퐹퐵 퐷퐹 ≠ 퐷퐹 퐹퐸 퐷퐹 퐶퐸 퐶퐵 퐶퐷 퐶퐹

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams Date: 9/5/14 S.119

5. Find . 6. Find .

푥 푥

7. Find . 8. Find .

푥 푥

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

퐷퐸 퐵퐶 퐷퐹 퐹퐸 퐵퐹 퐹퐶

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams Date: 9/5/14 S.120

10. In the circle shown, = 150°, = 30°, = 60°, = 8, = 4, = 12. a. Find . 푚퐷퐵퐺� 푚퐷퐵� 푚∠퐶퐸퐹 퐷퐹 퐷퐵 퐺퐹 b. Prove ~ . 푚∠퐺퐷퐵 c. Set up a proportion using sides and . ∆퐷퐵퐹 ∆퐸퐶퐹 d. Set up an equation with and using a theorem 퐶퐸�� 퐺퐸�� for segment lengths from this section. �퐶퐸�� 퐺퐸�� e. Solve for and .

퐶퐸 퐺퐸

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams Date: 9/5/14 S.121

Lesson 17: Writing the Equation for a Circle

Classwork Opening Exercise 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).

Lesson 17: Writing the Equation for a Circle Date: 9/5/14 S.122

Example 1

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

Lesson 17: Writing the Equation for a Circle Date: 9/5/14 S.123

Example 2

Shown below is a circle with center (2, 3) with radius 5.

Lesson 17: Writing the Equation for a Circle Date: 9/5/14 S.124

Exercises 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 푥 푦 −

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 푥 − 푦 −

Lesson 17: Writing the Equation for a Circle Date: 9/5/14 S.125

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? − 푥

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. 퐴 − 퐵

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

c. Write the equation of the circle.

Lesson 17: Writing the Equation for a Circle Date: 9/5/14 S.126

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

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.

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

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 Date: 9/5/14 S.127

Lesson Summary

( ) + ( ) = is the general equation for any circle with radius and center ( , ). 2 2 2 푥 − 푎 푦 − 푏 푟 푟 푎 푏

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 푥 푦 − 3. A circle is given by the equation + = 100. Which of the following points are on the circle? a. (0, 10) 2 2 푥 푦 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 푥 푦 c. 4( 2)2 + 4( 9)2 64 = 0 푥 푦 2 2 푥 − 푦 − − 5. A circle has center ( 13, ) and passes through the point (2, ). a. What is the radius of the circle? − 휋 휋 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? 퐴 퐵 − b. What is the radius of the circle? c. Write the equation of the circle.

Lesson 17: Writing the Equation for a Circle Date: 9/5/14 S.128

7. Write the equation of the circle shown below.

8. Write the equation of the circle shown below.

9. Consider the circles with equations: + = 2 and 2 2 ( 푥3) +푦( 3) = 32. 2 2 a. What are the radii of the two circles? 푥 − 푦 − 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 Date: 9/5/14 S.129

Lesson 18: Recognizing Equations of Circles

Classwork Opening Exercise

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

𝑥𝑥

3 3 𝑥𝑥 b. Express this as a trinomial: ( + 4) . 2 2 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 49 3 3 9 =

𝑥𝑥

c. Factor the trinomial : + 12 + 36. 2 𝑥𝑥 𝑥𝑥

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

Lesson 18: Recognizing Equations of Circles Date: 9/5/14 S.130

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 푦 − 푦

Exercise 1. Rewrite the following equations in the form ( ) + ( ) = . a. + 4 + 4 + 6 + 9 = 36 ퟐ ퟐ ퟐ 풙 − 풂 풚 − 풃 풓 2 2 𝑥𝑥 𝑥𝑥 푦 − 𝑥𝑥

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

Lesson 18: Recognizing Equations of Circles Date: 9/5/14 S.131

Example 2

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

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

b. 3 + 5 = 2 2 19 𝑥𝑥 − 𝑥𝑥 푦 − 푦 2

Lesson 18: Recognizing Equations of Circles Date: 9/5/14 S.132

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

4. Stella says the equation 8x + + 2y = 5 has a center of (4, 2) and a radius of 22. Is she correct? Why or why not? 2 2 𝑥𝑥 − 푦 − √

Example 3

Could + + + + = 0 represent a circle? 2 2 𝑥𝑥 푦 퐴𝑥𝑥 퐵푦 퐶

Lesson 18: Recognizing Equations of Circles Date: 9/5/14 S.133

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

b. + + 6 4 + 15 = 0 2 2 𝑥𝑥 푦 𝑥𝑥 − 푦

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

Lesson 18: Recognizing Equations of Circles Date: 9/5/14 S.134

Problem Set

1. Identify the center and radii of the following circles. a. ( + 25) + = 1 b. + 2 + 2 8 = 8 𝑥𝑥 푦 c. 2 20 + 2 10 + 25 = 0 𝑥𝑥 𝑥𝑥 푦 − 푦 d. 2 + = 192 𝑥𝑥 − 𝑥𝑥 푦 − 푦 2 2 e. 𝑥𝑥 + 푦 + + = 2 2 1 𝑥𝑥 𝑥𝑥 푦 푦 − 4 2. Sketch a graph of the following equations. a. + + 10 4 + 33 = 0 b. 2 + 2 + 14 16 + 104 = 0 𝑥𝑥 푦 𝑥𝑥 − 푦 c. 2 + 2 + 4 10 + 29 = 0 𝑥𝑥 푦 𝑥𝑥 − 푦 2 2 𝑥𝑥 푦 𝑥𝑥 − 푦 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 푦 − 𝑥𝑥 푦 − 𝑥𝑥 푦

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 . 퐴 퐴 퐴

Lesson 18: Recognizing Equations of Circles Date: 9/5/14 S.135

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 segment퐴 − ? 퐵 퐴퐵

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

c. What is the slope of ?

�퐴퐵��

d. Find the midpoint of .

�퐴퐵��

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

�퐴퐵��

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

Lesson 19: Equations for Tangent Lines to Circles Date: 9/5/14 S.136

Example 1

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

Lesson 19: Equations for Tangent Lines to Circles Date: 9/5/14 S.137

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

Lesson 19: Equations for Tangent Lines to Circles Date: 9/5/14 S.138

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 the2 upper2 half- plane?푝 푥 푦 푝 푥

Exercises 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 ? 푄′ ′ 푄

b. What is the slope of ?

�푂푄��′

c. What is the slope of ?

�푄′푃��

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

푝

Lesson 19: Equations for Tangent Lines to Circles Date: 9/5/14 S.139

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

4. Could a circle given by the equation ( 5) + ( 1) = 25 have tangent lines given by the equations 4 3 4 = ( 1) and 5 = ( 8)? Explain2 how 2you know. 3 4 푥 − 푦 − 푦 − 푥 − 푦 − 푥 −

Lesson 19: Equations for Tangent Lines to Circles Date: 9/5/14 S.140

Lesson Summary Theorems 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. 푥 − 푦 − b. Find the equations of the two tangent lines.

2. Consider the circle 4 + + 10 + 13 = 0. There are two lines tangent to this circle having a slope of . 2 2 2 a. Find the coordinates푥 − 푥 of the푦 two points푦 of tangency. 3 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 푥 푦 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 − − 푥 푦 5. What is the equation of the tangent line to the circle + = 1 through the point (6,0)? 2 2 푥 푦 6. D’Andre said that a circle with equation ( 2) + ( 7) = 13 has a tangent line represented by the equation 3 5 = ( + 1). Is he correct? Explain. 2 2 2 푥 − 푦 − 푦 − − 푥

Lesson 19: Equations for Tangent Lines to Circles Date: 9/5/14 S.141

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 ( + 1) + ( 9) = 145. 푦 − 푥 The circle2 has center2 ( 1,9) and radius 12. The point ( 10,1) is on the circle because 푥 푦 − ( 10 + 1) + (1 9) = ( 9) + ( 8) = 145. − − 2 2 2 2 8 The slope of the radius is = ; therefore, the equation of the tangent line is 1 = ( + 10). — − − − − 9 9−1 8 a. Kerry said that Kamal−1 has10 made9 an error. What was Kamal’s error? Explain what푦 − he did wrong.푥 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 푥 푥 푦 − 푦

Lesson 19: Equations for Tangent Lines to Circles Date: 9/5/14 S.142

Lesson 20: Cyclic Quadrilaterals

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

퐴퐵퐶퐷 푥 푦

Example 1:

Given quadrilateral with + = 180°, prove that quadrilateral is cyclic; in other words, prove that points , , , and lie on the same circle. 퐴퐵퐶퐷 푚∠퐴 푚∠퐶 퐴퐵퐶퐷 퐴 퐵 퐶 퐷

Lesson 20: Cyclic Quadrilaterals Date: 9/5/14 S.143

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. 퐷′′ 퐷′′

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

푃푄푅푆 △ 푃푄푇 △ 푆푅푇

Lesson 20: Cyclic Quadrilaterals Date: 9/5/14 S.144

3. A cyclic quadrilateral has perpendicular diagonals. What is the area of the quadrilateral in terms of , , , and as shown? 푎 푏 푐 푑

4. Show that the triangle in the diagram has area sin( ). 1 2 푎푏 푤

Lesson 20: Cyclic Quadrilaterals Date: 9/5/14 S.145

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

1 = ( + )( + ) sin( ) 6. Show that the area of the cyclic quadrilateral shown in the diagram is 2 . 퐴푟푒푎 푎 푏 푐 푑 푤

Lesson 20: Cyclic Quadrilaterals Date: 9/5/14 S.146

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 Area = sin( ). 2 푎 푏 푤 푎푏 ⋅ 푤 The area of a cyclic quadrilateral whose diagonals and intersect to form an acute or right angle with 1 degree measure : Area( ) = sin( ). 퐴퐵퐶퐷2 퐴퐶�� 퐵퐷�� 푤 퐴퐵퐶퐷 ⋅ 퐴퐶 ⋅ 퐵퐷 ⋅ 푤 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 .

퐵퐷퐶퐸 푂 푚∠퐵푂퐶 푚∠퐵퐸퐶

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

퐹퐴퐸퐷 퐴푋 퐹푋 푋퐷 푚∠퐴푋퐸 퐹퐴퐸퐷

Lesson 20: Cyclic Quadrilaterals Date: 9/5/14 S.147

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

�퐵퐸�� ∥ �퐶퐷�� 푚∠퐵퐸퐷 푠 푡

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

�퐵퐶�� 푚∠퐵퐶퐷 퐶퐸�� ≅ �퐷퐸�� 푚∠퐶퐸퐷

5. In circle , = 15°. Find .

퐴 푚∠퐴퐵퐷 푚∠퐵퐶퐷

Lesson 20: Cyclic Quadrilaterals Date: 9/5/14 S.148

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

푂 푚∠푁푂푃 푚∠푃푄퐸

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

퐶 퐵 퐸 퐷

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

퐽퐾퐿푀 푛

Lesson 20: Cyclic Quadrilaterals Date: 9/5/14 S.149

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 . 퐴 퐵 푚∠퐹퐻퐺 푚∠퐻퐺퐸 푚∠퐺퐸퐹 푚∠퐸퐹퐻

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 quadrilateral? 푎 푏 푐 푑

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. 푃푆푅푄 �푃푄�� ∠푄푅푇 ≅ ∠푄푆푅 ∠푃푇푅 푆 푋 푇 푃

Lesson 20: Cyclic Quadrilaterals Date: 9/5/14 S.150

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 cyclic퐴퐵퐶퐷 quadrilateral.퐴퐶 ⋅ 퐵퐷 Label퐴퐵 its⋅ 퐶퐷 vertices퐵퐶 ⋅,퐴퐷 , , and . Draw the two diagonals and . 퐴 퐵 퐶 퐷 �퐴퐶�� 퐵퐷��

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

Repeat for a second example of a cyclic quadrilateral.퐴퐶 ⋅ 퐵퐷 퐴퐵 ⋅ 퐶퐷 퐵퐶 ⋅ 퐴퐷 Challenge: Draw a cyclic quadrilateral with one side of length zero. What shape is the this cyclic quadrilateral? Does Ptolemy’s claim hold true for it?

Lesson 21: Ptolemy’s Theorem Date: 9/5/14 S.151

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 = . 퐴퐵퐶퐷 퐴퐶 퐵퐷 푤 퐴퐵 푎 퐵퐶 푏 퐶퐷 a.푐 From퐷퐴 last 푑lesson, what is the area of quadrilateral in terms of the lengths of its diagonals and the angle ? Remember this formula for later on! 퐴퐵퐶퐷 푤

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

∠퐵퐶퐷 ∠퐵퐴퐷

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∠퐵퐶퐷 is the area of quadrilateral in terms of , , , , and ? 푣 퐵퐶퐷 푏 푐 푣 퐵퐴퐷 푎 푑 푣 퐴퐵퐶퐷 푎 푏 푐 푑 푣

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? 퐴퐵퐶퐷

Lesson 21: Ptolemy’s Theorem Date: 9/5/14 S.152

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. 퐴 퐵 퐶 퐷 퐴퐵퐶 i. What does the reflection do with points and ? �퐴퐶�� 퐴′ 퐵′ 퐶′ 퐴 퐵 퐶 퐴 퐶

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

퐵′

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

퐴퐵′퐶퐷 퐴퐵퐶퐷

f. The diagram shows angles having degree measures , , , , and . Find and label any other angles having degree measures , , , , or , and justify your answers. 푢 푤 푥 푦 푧 푢 푤 푥 푦 푧

Lesson 21: Ptolemy’s Theorem Date: 9/5/14 S.153

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

푤 푢 푧

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

푢 푥 푦 푧 푤 퐴퐵′퐶퐷

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 . ′ ′ 퐵 퐴퐷 푏 푑 푤 퐵 퐶퐷 푎 푐 푤

j. Based on the results of part (i), write a formula for the area of cyclic quadrilateral In terms of , , , , and . 퐴퐵퐶퐷 푎 푏 푐 푑 푤

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

Lesson 21: Ptolemy’s Theorem Date: 9/5/14 S.154

Lesson Summary

Theorems

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

퐴퐵퐶퐷 퐴퐶 ⋅ 퐵퐷 퐴퐵 ⋅ 퐶퐷 퐵퐶 ⋅ 퐴퐷 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? 푃

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 perimeter퐴퐵퐶퐷 of the kite? ∶

Lesson 21: Ptolemy’s Theorem Date: 9/5/14 S.155

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 about푎 this rectangle?푏 푐

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? 푏

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

Lesson 21: Ptolemy’s Theorem Date: 9/5/14 S.156

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 exact푥 value푦 of sin(75°). 푎 푏 푐 푑 푥 푦

a. Explain why ( ) equals the diameter of the circle. 푎 b. If the circle hassin a푥 diameter of 1, what is ? c. Use Thales’ theorem to write the side lengths in the original diagram in 푎 terms of and . d. If one diagonal of the cyclic quadrilateral is 1, what is the other? 푥 푦 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 Date: 9/5/14 S.157