Download - P DIAMETER: Distance across the circle through its center Also known as the longest chord
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P
DIAMETER:Distance across the circle through its centerAlso known as the longest chord.
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P
RADIUS:
Distance from the center to point on circle
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Formula
Radius = ½ diameteror
Diameter = 2r
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D = ?
r = ?
r = ? D = ?
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Secant Line:intersects the circle at exactly TWO points
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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
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Name the term that best describes the notation.
Secant
Radius
DiameterChord
Tangent
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Central Angles
An angle whose vertex is at the center of the circle
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P
E
F
D
Semicircle: An Arc that equals 180°
EDF
To name: use 3 letters
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THINGS TO KNOW AND REMEMBER ALWAYS
A circle has 360 degrees
A semicircle has 180 degrees
Vertical Angles are CONGRUENT
Linear Pairs are SUPPLEMENTARY
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Formulameasure Arc = measure
Central Angle
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m AB
m ACB
m AE
A
B
C
Q96
E=
=
=
96°
264°
84°
Find the measures. EB is a diameter.
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Tell me the measure of the following arcs.
AC is a diameter.
80
10040
140A
B
C
D
Rm DAB =
m BCA =
240
260
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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.
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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
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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
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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.
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The circle graph shows the types of cuisine available in a city. Find mTRQ.
158.4
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Inscribed Angle
Inscribed Angle = intercepted Arc/2
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160
80
The inscribed angle is half of the intercepted angle
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120
x
y
Find the value of x and y.
= 120
= 60
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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
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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
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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
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1. Solve for arc ABC
2. Solve for x and y.
244
x = 105y = 100
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Vertex is INSIDE the Circle NOT at the Center
Arc+ArcANGLE =
2
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Ex. 1 Solve for x
X
8884
x = 100
180 – 88
92
8492
2x
184 84 x
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Ex. 2 Solve for x.
45
93
xº
89x = 89
360 – 89 – 93 – 45
133
133 452
x
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Vertex is OUTside the Circle
Large Arc Small ArcANGLE =
2
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x
Ex. 3 Solve for x.
65°
15°
x = 25
65 152
x
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x
Ex. 4 Solve for x.
27°
70°
x = 16
7027
2x
54 70 x
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x
Ex. 5 Solve for x.
260°
x = 80
360 – 260
100
260 1002
x
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Warm up: Solve for x
18◦
1.)
x
124◦
70◦
x
2.)
3.)
x
260◦
20◦110◦ x
4.)
53 145
8070
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Circumference, Arc Length, Area, and Area of Sectors
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Find the EXACT circumference.
28 ftC 1. r = 14 feet
2. d = 15 miles
15 milesC
2 14C
15C
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Ex 3 and 4: Find the circumference. Round to the nearest tenths.
89.8 mmC 103.7 ydC
2 14.3C 33C
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Arc LengthThe distance along the curved line
making the arc (NOT a degree amount)
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Arc Length
measure of arc
Arc Length 2360
r
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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
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Ex 6. Find the exact Arc Length.
Arc Length 10
in3
=
measure of arc
Arc Length 2360
r
120Arc Length 2 5
360
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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
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Area of CirclesThe amount of space occupied.
r A = pr2
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Find the EXACT area.
2841 ftA 8. r = 29 feet
9. d = 44 miles
2484 miA
229A
2442
A
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10 and 11Find the area. Round to the nearest tenths.
2181.5 ydA 22206.2 cmA
27.6A
2532
A
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Area of a Sectorthe region bounded by two radii of the
circle and their intercepted arc.
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Area of a Sector
2measure of arc
360A r
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Example 12Find the area of the sector to the nearest hundredths.
A 18.85 cm2
606 cm
Q
R
2606
360A
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Example 13 Find the exact area of the sector.
6 cm
120
7 cm
Q
R
249A cm
3
21207
360A
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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