Chapter 9Circles

• Define a circle and a sphere.

• Apply the theorems that relate tangents, chords and radii.

• Define and apply the properties of central angles and arcs.

With your compass…

• Create a circle on your notes that is about half the page

• As we go through the basic terms for a circle label each one on your notes

9.1 Basic Terms

Objectives

• Define and apply the terms that describe a circle.

The Circle

is a set of points in a plane at a given distance from a given point in that plane

A

B

The Circle

The given distance is a radius (plural radii)

A

B

radius

How many radii does a circle have?

Are they all the same length?

The Circle

The given point is the center

A

B

radius

center “Circle with center A”

A

The Circle

A

BPoint on circle

What are some real world examples of circles?

Think – Pair - ShareThink – Pair - Share

•What is the difference between

a line and a line segment?

Chord

any segment whose endpoints are on the circle.

A

BC

chord

Diameter

A chord that contains the center of the circle

A

B

C

diameterWhat is another name for half of the diameter?

any line that contains a chord of a circle.

Secant

A

BC

secant

Tangent

any line that contains exactly one point on the circle.

A

B

tangent

C

ABC is tangent to

Point of Tangency

A

BPoint of tangency

Common Tangentsare lines tangent to more than one coplanar circle.

A

XB

R

X

B

Common External Tangents

A

X

B

R

D

Y

Common Internal Tangents

Tangent Circles

are circles that are tangent to each other.

A

B

R

A

B

R

Internally Tangent Circles

Externally Tangent Circles

Sphere

is the set of all points in space equidistant from a given point.

AB

Sphere

Radii

Diameter

Chord

Secant

TangentA

B

D

C

E

F

Congruent Circles (or Spheres)

WHAT DO THEY HAVE?• have equal radii.

A D

BE

Concentric Circles (or Spheres)

share the same center.

O

G

Q

Who can think of a real world example?

Think , throwing of pointy objects….

Inscribed/Circumscribed

A polygon is inscribed in a circle and the circle is circumscribed about the polygon if each vertex of the polygon lies on the circle.

P

M

Q

O

N

R

L

Name…1. 3 radii2. Diameter3. Chord 4. Secant 5. Tangent6. What is the name for ZX?7. What should point M be called?

X

Z

White Boards….

• Draw the following…1. An inscribed triangle

2. A circle circumscribed about a quadrilateral

3. 2 circles with common external tangents

4. 2 circle that are internally tangent to each other

5.

9.2 Tangents

Objectives

• Apply the theorems that relate tangents and radii

Experiment

• Supplies: Pencil, protractor, compass 1. Draw a circle with center X

2. Draw Point Y on the bottom of your circle

3. Create line ZY tangent to the circle at Point Y

4. Draw the radius to the point of tangency and measure the angle formed by the tangent and the radius (L XYZ)

5. Compare your measurements with those around you…

TheoremIf a line is tangent to a circle, then the line is perpendicular to the radius

drawn to the point of tangency.

A

B

tangent

C

90m ABC

What can we conclude based on our experiment?

Theorem (fill in the blank)

If a line in the plane of a circle is perpendicular to a radius at its endpoint,

AX

B

tangent

then the line is a tangent to the circle.

Dunce Cap Rule

Tangents to a circle from a common point are congruent.

Y Z

A

X

tangent

tangent

How do we know the 2 right triangles are congruent?

Inscribed/Circumscribed

When each side of a polygon is tangent to a circle, the polygon is said to be circumscribed about the circle and the circle is inscribed in the polygon.

Each side of the poly, is what to the circle?

• GIVEN– Radius = 6– BC = 8 – Find AC

• What allows you to come up with the correct answer?

• GIVEN– LC = 45– BC = 4 – Find AC

A

B C

BC is tangent to A

Whiteboards

• Create a diagram of the following…1. A triangle circumscribed about a circle

2. A pentagon inscribed inside a circle

9.3 Arcs and Central Angles

Objectives

• Define and apply the properties of arcs and central angles.

Experiment

1. Draw a straight line using a straightedge

2. Extend the line so that its sides go past the outside of your protractor

3. Connect the two sides of your angle using the outer curved side of your protractor

4. What do you end up with?5. What do I get with a 70

degree angle?

Think – pair - shareThink – pair - share

How is the angle measurement that you just created related to the measurement of a

circle?

Central Angle

is formed by two radii, with the center of the circle as the vertex.

B

A

C

It’s like cutting out a slice of pizza!!

This represents the crust of your pizza

Arcan arc is part of a circle

like a segment is part of a line.

AC

ABCL

Central Angle / Arc Measurethe measure of an arc is given by the measure of its

central angle. (or vise versa)

B

AC

80

80

80mAC

The central angle tells us how much of the 360 ◦ of crust we are using from our pizza.

ABCm L = 80

Minor Arc

an unbroken part of a circle with a measure less than 180°.

BA

C

AC

Semicircle

an unbroken part of a circle that shares endpoints with a diameter.

B

A C

ABC

How do I know that AC is a diameter?

Major Arcan unbroken part of a circle with a measure greater

than 180°.

BA C

D

ACD

We only know how to measure angles up to 180, so how do you find the measure of a major arc?

Practice

Name two minor arcs

R

C

SA

O

AR, RC, RS, AS, SC

Practice

Name two major arcs

R

C

SA

O

White Board Practice

Name two semicircles

R

C

SA

O

Skip Remember….

• Adjacent angles ?• Angle addition postulate?

• Smartboard

Skip Adjacent Arcs

arcs that have exactly one point in common.

B

A C

D

AD DC

Arc Addition Postulate

B

A

C

D

mADCmDCmAD

Theorem

In the same circle or in congruent circles….

• congruent arcs = congruent central angles

B

A C

D

90 90DCAD

DBCABD

90 90

smartboard

Dissecting a Circle Diagram

FREE VISUAL EVIDENCE !!!• Central angles = minor arcs• All the arcs = 360• Diameters = straight lines = 180• Vertical angles / adjacent angles

R

C

SA

O

H

White Board Practice

R

C

SA

O

H

30

50

Find the following measurements…

RH

AOR

HCA

100

50

210

Group Practice

• Give the measure of each arc.

4x

3x 3x + 10

2x

2x-1

4

A

B

C

D

E

Group Practice

m AB = 88

m BC = 52

m CD = 38

m DE = 104

m EA = 784x

3x 3x + 10

2x

2x-1

4

A

B

C

D

E

The radius of the Earth is about 6400 km.

6400

6400

O

BA

The latitude of the Arctic Circle is 66.6º North. That means the m BE 66.6º.

6400

6400

O

BA

EW

66.6º

Find the radius of the Arctic Circle

6400

O

BA

EW

66.6º

xº

Find the radius of the Arctic Circle

6400

O

BA

EW

66.6º

23.4º

Section 9-4 Arcs and Chords

Objectives

• Define the relationships between arcs and chords.

REVIEW

• WHAT IS A CHORD?

• WHAT IS AN ARC?

Relationship between a chord and an arc

The minor arc between the endpoints of a chord is called the arc of the chord, and the chord between the endpoints of an arc is the chord of the arc.

BA

D

Chord BD “cuts off” 2 arcs..

BD and BFDF

Theorem In the same circle or in congruent circles…• congruent arcs have congruent chords • congruent chords have congruent arcs.

BA

C

D

F.

If arc BD is congruent to arc FC then…

Skip - Midpoint/ Bisector of an Arc

• Just as we have learned about the bisectors and midpoints of angles and line segments, an arc can be bisected into two congruent arcs

B AC

DIf D is the midpoint of arc BDC, then…….

Circle Handout Experiment

1. Label the center A

2. DRAW A CHORD AND LABEL IT DC

3. FIND THE MIDPOINT OF THE CHORD AND LABEL IT X

4. DRAW A RADIUS THAT PASSES THROUGH THE MIDPOINT AND INTERSECTS THE ARC OF THE CIRCLE AT Y

5. USE A PROTRACTOR AND MEASURE LAXC

Think – Pair - ShareThink – Pair - Share

• What facts can you conclude about the arcs, chords, or any other segments?

• What is congruent to what?

• What about perpendicular?

Theorem A diameter (or radius) that is perpendicular to a chord

bisects the chord and its arc.

B

AC

D

X

Y

DC BY

DX XC

DY YC

EC: What other 2 segments do I know are congruent that are not explicitly shown?

Remember

• When you measure distance from a point to a line, you have to make a perpendicular line.

A

Putting Pythagorean to Work..

BA

C

D

X

Y

E

AY = 3 AX = 3

4

Partners: Use the given information do make a conclusion about the chords shown.

Hint: Just because something is not shown, doesn’t mean it doesn’t exist! (other radii)

Theorem • In the same circle or in congruent circles:

– Chords equally distant from the center (or centers) are congruent

– Congruent chords are equally distant from the center (or centers)

.

B

AC

D

X

YA XA

BD EC

Y

E

WHITEBOARDS

B

A

5

D

13

yxC

Solve for x and y

x = 12y = 12

IF arc DB is 55 degrees, then arc CB is?

WHITEBOARDS

yx

Solve for x and y

x = 8y = 16 8

Whiteboards

• Find the length of a chord 3cm from the center of a circle with a radius of 7cm.

WARM - UP

1. What does the term inscribed mean to you in your own words?

– Describe the placement of the vertices of an inscribed triangle

2. What do we call the 2 sides and vertex (in circle terms) of a central angle that you learned in 9.3?

– What is the measure of the angle equal to?

9.5 Inscribed Angles

Objectives• Solve problems and

prove statements about inscribed angles.

• Solve problems and prove statements about angles formed by chords, secants and tangents.

What we’ve learned…• Inscribed means that something is inside of

something else – we have looked at inscribed polys and circles

• We know that an angle by definition has a vertex and 2 sides that meet at the vertex

• In a central angle…– The vertex is the center and the sides are radii

Inscribed Angle

B

A

C

An angle formed by two chords or secant lines whose vertex lies on the circle.

“Intercepted arc”

What do you think the sides are in ‘circle

terms’? Where is the vertex?

Create a circle using a compass & an inscribed angle in your

notes

Experiment

1. Measure the inscribed angle created with a protractor

2. Using the endpoints of the intercepted arc, draw 2 radii to create a central angle and then measure.

3. Compare the measurement of the inscribed angle with that of the central angle measure.

4. Discuss with your partner

Theorem

B

A

C

The measure of an inscribed angle is half the measure of the intercepted arc.

mACABCm ___2

Corollary

B

A

C

If two inscribed angles intercept the same arc, then they are congruent.

ABC ADC D

Don’t write down, just recgonize

WHITEBOARDS• Find the values of r, s, x, y , and z

– Take inventory of the diagram before trying to solve!

– Concentrate on parts of the whole

z◦

y◦

x◦

O 80

r = 50s = 50x = 160y = 100z = 100

r◦

s◦

Corollary

C

A

If the intercepted arc is a semicircle, the inscribed angle must = 90.

90m ABC

B

O

What is the measure of an inscribed angle whose intercepted arc has the endpoints of the diameter?

Corollary

C

A

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

B

O D

180

180

m A m C

m B m D

Theorem

C

A

B

O

D

mADBABCm

F

Treat this angle the same as you would and inscribed angle!

___2

Whiteboards

• Page 353 – #7, 6

WHITEBOARDS

z◦

60◦

x◦

O

y◦

• Find the values of x, y , and z

X = 30Y = 60Z = 150

9.6 Other Angles

Objectives

• Solve problems and prove statements involving angles formed by chords, secants and tangents.

12

3

4

WARM UP: •Draw a central angle and label it •Draw an inscribed angle and label it•FOR BOTH CREATE AN EQUATION FOR THEIR MEASURES IN COMPARISON TO THE INTERCEPTED ARC

WARM UP: •For this diagram….•Write down the different equations that represent the angle relationships shown.•There’s more than one!

TheoremThe angle formed by two intersecting chords

is equal to half the sum of the intercepted arcs.

A

D

B

C

E

1

2

Partners: How do you think we can determine the measure of L1?

)(1 mDEmCBm _________2

)(2 mDBmCEm _________2

Angles formed from a point outside the circle…

2 secants2 tangents

1tangent1 secant

In each circle, 2 arcs are being intercepted by the angle.

The larger arc is always further away from the vertex.

TheoremThe angle formed by secants or tangents with the

vertex outside the circle has a measure equal to half the difference of the intercepted arcs.

A

D

B

CE

1

F

)(1 mEFmBDm _________2

WHITEBOARDS

• ONE PARTNER OPEN BOOK TO PG. 358

• ANSWER #1– 35

• ANSWER # 6– 40

• ANSWER # 4– 80

• ANSWER # 7– X=50

AO

G

F

D

E

CB

123

45

6

7

8

AB is tangent to circle O.AF is a diameterm AG = 100m CE = 30m EF = 25

100

30

25

WARM – UP

• READ PG. 361 – 363– Identify what elements are involved in each of

the 3 theorems in this section – Example: “Theorem 9-11 refers to the

relationship of 2 ___________ intersecting”

– What is the idea behind this section…. Angles, segments, circles?

9.7 Circles and Lengths of Segments

Objectives

• Solve problems about the lengths of chords, secants and tangents.

Skip - Open Books to Pg. 361

• Read the paragraphs for section 9-7

• “segments of a chord”– AP and PB

O

B

D

P

A

C

TheoremIf a chord intersects another

chord each chord now has how many parts?– If you multiply those 2 parts

together they will equal the product of the other chords 2 parts.

D

B

X

E

F

XBFXXDEX

Example

2

3

12

x

Solve for x

2 • 12 = 3 • x 24 = 3x 8 = x

Skip - Pg. 362• Read the middle

paragraph

• “external segments”– aka “outside piece”– BP and DP

• “Secant segments”– Aka “whole piece”– AP and CP

O

BD

P

A

C

Theorem

A

D

B

CE

FCFCDCECB

Whole Outside Part = Whole Outside Part

2 Secants •Each one is made up of an outside part and an inside part that make the whole.

Example

8

8

x

4

8 •x = 12 •48x = 48x = 6

Whole Outside Part = Whole Outside Part

Skip Pg. 362 and 363

• Read the bottom of 362 and top of 363

• “external segments”– BP and PD

• “Secant segments”– AP and PD

A

P

D

B

Whole Outside Part = Whole Outside Part

Theorem

A

D

C

E

FCECECFCD

Whole Outside Part = Whole Outside Part

1 Secant •Still the same parts… outside + inside = whole

1 Tangent •Only made up of an outside part…which is also the whole!!

Example

186

x

Whole Piece Outside Piece = Whole Piece Outside Piece

x •x = 24 •6x2 = 144x = 12

WHITEBOARDS• ONE PARTNER OPEN BOOK TO PG. 363/364

• ANSWER #1– x = 8

• ANSWER # 4– x = 4

• ANSWER # 5– x = 4

x

5

4

y

3

WHITEBOARDS

X = 9Y = 6

x 5

4

1

3A

B

CD

E

FG

H

I

7 2

6

ANGLES QUIZ

1. Central angle2. Inscribed angle3. Angle formed inside4. Angle formed outside5. 90 angle

8

• Identify a numbered angle that represents each of the bullet points.

• Write an equation representing the measure of the angle •i.e. m L12 =

K