P

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

P

RADIUS:

Distance from the center to point on circle

Formula

Radius = ½ diameteror

Diameter = 2r

D = ?

r = ?

r = ? D = ?

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 one radius

Point of Tangency: The point where the tangent intersects the circle

Name the term that best describes the notation.

Secant

Radius

DiameterChord

Tangent

Central Angles

An angle whose vertex is at the center of the circle

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 degrees

A semicircle has 180 degrees

Vertical Angles are CONGRUENT

Linear Pairs are SUPPLEMENTARY

Formulameasure Arc = measure

Central Angle

m AB

m ACB

m AE

A

B

C

Q96

E=

=

=

96°

264°

84°

Find the measures. EB is a diameter.

Tell me the measure of the following arcs.

AC is a diameter.

80

10040

140A

B

C

D

Rm DAB =

m BCA =

240

260

Using Properties of Tangents

HK and HG are tangent to F. Find HG.

HK = HG

5a – 32 = 4 + 2a

3a – 32 = 4

2 segments tangent to from same ext. point segments .

Substitute 5a – 32 for HK and 4 + 2a for HG.

Subtract 2a from both sides.

3a = 36

a = 12

HG = 4 + 2(12)

= 28

Add 32 to both sides.

Divide both sides by 3.

Substitute 12 for a.

Simplify.

Applying Congruent Angles, Arcs, and Chords

TV WS. Find mWS.

9n – 11 = 7n + 11

2n = 22

n = 11

= 88°

chords have arcs.

Def. of arcs

Substitute the given measures.

Subtract 7n and add 11 to both sides.

Divide both sides by 2.

Substitute 11 for n.

Simplify.

mTV = mWS

mWS = 7(11) + 11

TV WS

Example 3B: Applying Congruent Angles, Arcs, and Chords

C J, and mGCD mNJM. Find NM.

GD = NM

arcs have chords.GD NM

GD NM GCD NJM

Def. of chords

Find QR to the nearest tenth.

Step 2 Use the Pythagorean Theorem.

Step 3 Find QR.

PQ = 20 Radii of a are .

TQ2 + PT2 = PQ2

TQ2 + 102 = 202

TQ2 = 300TQ 17.3

QR = 2(17.3) = 34.6

Substitute 10 for PT and 20 for PQ.Subtract 102 from both sides.Take the square root of both sides.

PS QR , so PS bisects QR.

Step 1 Draw radius PQ.

The circle graph shows the types of cuisine available in a city. Find mTRQ.

158.4

Inscribed Angle

Inscribed Angle = intercepted Arc/2

160

80

The inscribed angle is half of the intercepted angle

120

x

y

Find the value of x and y.

= 120

= 60

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

3

Q

D

JT

U

4

5x = 2x + 9

x = 3

3x = + 9

4x – 14 = 90

H

K

GN

Example 4

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

x = 26

4x = 104

z

2x + 18

85

2x +18 + 22x – 6 = 180

x = 7

z + 85 = 180z = 95

Example 5 Solve for x and z.

22x – 6

24x +12 = 18024x = 168

1. Solve for arc ABC

2. Solve for x and y.

244

x = 105y = 100

Vertex is INSIDE the Circle NOT at the Center

Arc+ArcANGLE =

2

Ex. 1 Solve for x

X

8884

x = 100

180 – 88

92

8492

2x

184 84 x

Ex. 2 Solve for x.

45

93

xº

89x = 89

360 – 89 – 93 – 45

133

133 452

x

Vertex is OUTside the Circle

Large Arc Small ArcANGLE =

2

x

Ex. 3 Solve for x.

65°

15°

x = 25

65 152

x

x

Ex. 4 Solve for x.

27°

70°

x = 16

7027

2x

54 70 x

x

Ex. 5 Solve for x.

260°

x = 80

360 – 260

100

260 1002

x

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, Area, and Area of Sectors

Find the EXACT circumference.

28 ftC 1. r = 14 feet

2. d = 15 miles

15 milesC

2 14C

15C

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

89.8 mmC 103.7 ydC

2 14.3C 33C

Arc LengthThe distance along the curved line

making the arc (NOT a degree amount)

Arc Length

measure of arc

Arc Length 2360

r

Ex 5. Find the Arc LengthRound to the nearest hundredths

8m

70

Arc Length 9.7= 7 m

measure of arc

Arc Length 2360

r

70Arc Length 2 8

360

Ex 6. Find the exact Arc Length.

Arc Length 10

in3

=

measure of arc

Arc Length 2360

r

120Arc Length 2 5

360

Ex 7. What happens to the arc length if the radius were to be doubled? Halved?

20Doubled

35

Halved 3

measure of arc

Arc Length 2360

r

Area of CirclesThe amount of space occupied.

r A = pr2

Find the EXACT area.

2841 ftA 8. r = 29 feet

9. d = 44 miles

2484 miA

229A

2442

A

10 and 11Find the area. Round to the nearest tenths.

2181.5 ydA 22206.2 cmA

27.6A

2532

A

Area of a Sectorthe region bounded by two radii of the

circle and their intercepted arc.

Area of a Sector

2measure of arc

360A r

Example 12Find the area of the sector to the nearest hundredths.

A 18.85 cm2

606 cm

Q

R

2606

360A

Example 13 Find the exact area of the sector.

6 cm

120

7 cm

Q

R

249A cm

3

21207

360A

Area of minor segment =

(Area of sector) – (Area of triangle)

12 yd

2 1

Area of minor segment =360 2

mRQr b hR

Q290 1

= (12) (12)(12)360 2

=113.10 722Area of minor segment =41.10yd

Example 14