Unit 3 Circles

Parts of a CircleCircle – set of all points _________ from a given point called the _____ of the circle.

C

Symbol:

equidistant

center

C

CHORD:A segment whose endpoints are on the circle

P

RADIUS:Distance from the center to point on circle

Radius

Diameter

P

DIAMETER:Distance across the circle through its centerAlso known as the longest chord.

D = ?

r = ?

r = ? D = ?

Use P to determine whether each statement is true or false.

P

QR

TS

diameter. a is .1 RT False

radius. a is .2 PS True

chord. a is .3 QT True

Secant Line:intersects the circle at exactly TWO points

a LINE that intersects the circle exactly

ONE time

Tangent Line:

Forms a 90°angle with a radiusPoint of Tangency: The point where the tangent intersects the circle

Name the term that best describes the notation.

Secant

Radius

DiameterChordTangent

P

A

BC

Central Angle : An Angle whose vertex is at the center of the

circleMinor ArcMajor Arc

Less than 180°

More than 180°

ABACBTo name:

use 2 letters

To name: use 3 letters

APB is a Central Angle

P

E

F

D

Semicircle: An Arc that equals 180°

EDF

To name: use 3 letters

THINGS TO KNOW AND REMEMBER ALWAYS

A circle has 360 degreesA semicircle has 180

degreesVertical Angles are Equal

measure of an arc = measure of central angle

A

B

C

Q 96

m AB

m ACBm AE

E

=

==

96°

264°84°

Arc Addition PostulateA

BC

m ABC =

m AB + m BC

Tell me the measure of the following arcs.

80100

40

140A

B

C

DR

m DAB =m BCA =

240

260

Congruent Arcs have the same measure and MUST come from the same circle or of congruent circles.

4545

A

BC

D

110

Arc length is proportional to “r”

Warm up

Central Angle

Angle = Arc

Inscribed Angle• Angle where the vertex in

ON the circle

Inscribed Angle

ARCANGLE = 2

2 ArcdIntercepteAngleInscribed

160

80

The arc is twice as big as the angle!!

120

x

y

Find the value of x and y.

Examples1. If mJK = 80 and JMK = 2x – 4, find x.

M

Q

K

S

J

2. If mMKS = 56, find m MS.x = 22

112

72˚

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

Find the measure of DOG and DIG D

O

G

I

If all the vertices of a polygon touch the edge of the circle, the polygon is INSCRIBED and the circle is CIRCUMSCRIBED.

Quadrilateral inscribed in a circle: opposite angles are SUPPLEMENTARY

A B

CD

180 CmAm180 DmBm

If a right triangle is inscribed in a circle then the hypotenuse is the diameter of the circle.

diameter

Example 3In J, m3 = 5x and m 4 = 2x + 9.Find the value of x.

3

QD

JT

U

4x = 3

4x – 14 = 90

HK

GN

Example 4In K, GH is a diameter and mGNH = 4x – 14. Find the value of x.

x = 26

Bonus: What type of triangle is this? Why?

z

y

110

85

110 + y =180y = 70

z + 85 = 180z = 95

Example 5 Find y and z.

Warm Up

1. Solve for arc ABC

2. Solve for x and y.

244

x = 105y = 100

Wheel of Formulas!!

Vertex is INSIDE the Circle NOT at the Center

Arc+ArcANGLE = 2

Ex. 1 Solve for x

X

8884

x = 100

Ex. 2 Solve for x.

45

93

xº

89x = 89

Vertex is OUTside the Circle

Large Arc Small ArcANGLE = 2

x

Ex. 3 Solve for x.

65°

15°

x = 25

x

Ex. 4 Solve for x.

27°

70°

x = 16

x

Ex. 5 Solve for x.

260°

x = 80

Tune: If You’re Happy and You Know It

• If the vertex is ON the circle half the arc. <clap, clap>

• If the vertex is INside the circle half the sum. <clap, clap>

• But if the vertex is OUTside, then you’re in for a ride, cause it’s half of the difference anyway. <clap, clap>

Warm up: Solve for x

18◦

1.)

x

124◦70◦

x

2.)

3.)

x

260◦

20◦110◦ x

4.)

53 145

8070

Circumference & Arc Length

of Circles

2 Types of AnswersRounded

• Type the Pi button on your calculator

• Toggle your answer

• Round

Exact• Type the Pi

button on your calculator

• Pi will be in your answer

• TI 36X Pro gives exact answers

CircumferenceThe distance around a circle

orC 2 r

C d

Circumference

Find the EXACT circumference.

28 ft1. r = 14 feet

2. d = 15 miles 15 miles

C 2 14

C 15

Ex 3 and 4: Find the circumference. Round to the nearest tenth.

89.8 mm 103.7 ydC 2 14.3 C 33

5. A circular flower garden has a radius of 3 feet. Find the circumference of the garden to the nearest hundredths.

C = 18.85 ft

r2 C C 2 3

Arc LengthThe distance along the curved line

making the arc (NOT a degree amount)

Arc Length

measure of arcArc Length 2

360r

Ex 6. Find the Arc LengthRound to the nearest hundredths

8m

70

Arc Length 9.77 m

measure of arcArc Length 2

360r

70

Arc Length 2 8360

Ex 7. Find the exact Arc Length.

10Arc Length in3

measure of arcArc Length 2

360r

120

Arc Length 2 5360

Ex 8 Find the radius. Round to the nearest hundredth.

B

A

»Arc Length of 3.82AB m=

60◦

3.65 mr

60

3.82 2360

r

1375.2 60 2 r

1375.2 120 r11.46 r

Ex 9 Find the circumference. Round to the nearest hundredth.

80◦

8032.11 2

360r

11,559.6 80 2 r

144.50inC

»Arc Length of 32.11inAB =

B

A

11,559.6 80 C

Ex 10 Find the radius of the unshaded region. Round to the nearest tenth.

75◦ B

A

»Arc Length of 10AB cm=

75

10 2360

r

3600 75 2 r3600 150 r

24 r7.6r cm