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