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Circles
2
3
Definitions
• A circle is the set of all points in a plane that are the same distance from a fixed point called the center of the circle.
• A radius of a circle is a line segment extending from the center to the circle.
• A diameter is a line segment that joins two points on the circle and passes through the center.
radius
center
diameter
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Naming a Circle
• A circle in a diagram is named by its center. The circle at right is called circle O or:
• If there is more than one circle in a diagram with the same center, this notation does not suffice.
• Note: two circles in the same plane with the same center are called concentric circles.
O
O
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• The word radius (plural: radii) is also used to denote the length of a radius (all radii have the same length).
• The word diameter is also used to denote the length of a diameter (all diameters have the same length).
• Note that the diameter of a circle is twice its radius.
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Chords
• A chord is any line segment that joins two points on a circle.
• Therefore, a diameter is an example of a chord. It is the longest possible chord.
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Chords and Radii
• Given a chord in a circle, any radius that bisects the chord (passes through its midpoint) is perpendicular to that chord.
• Also, if a radius is perpendicular to a chord, then it bisects the chord.
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Distance to Chords
• The distance from the center of a circle to a chord is measured along the radius that is perpendicular to the chord.
• Chords that are the same distance from the center are the same length.
• Also, chords that are the same length are the same distance from the center.
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Example
• In a circle of radius 5, a chord has length 8. Find the distance to the chord from the center of the circle.
• Let C be the center of the circle and A and B the endpoints of the chord. Let M be the midpoint of the chord so that
• Then • So, by the Pythagorean Theorem,
CA
B
M
.CM AB5 and 4.AC AM
5
4
2 25 4 25 16 9 3.MC
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Example
• In the figure,
• What is
• Since are both radii, they are congruent.
• So, angles A and B are congruent too. Therefore,
P
A
B
20 .m A ?m P
and AP BP
180 2(20 ) 140 .m P
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Intersecting Chords
• Consider the two intersecting chords in the figure. Each chord is cut into two pieces.
• The product of the lengths of the two pieces on one chord equals the product of the lengths of the two pieces on the other chord:
A
B
C
D
P
AP PB CP PD
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Example
• In the figure,
• If then find
• Let denote the length of CP.
• Then is the length of PD.
• Then
A
B
C
D
P
3, 6, and 11.AP PB CD .CP
3
6x
11-x11 x
x
(11 ) 3 6x x 211 18x x
2 11 18 0x x
,CP PD
( 9)( 2) 0x x 9 or 2x x So, 9.CP
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Tangents
• Given a circle, a line is tangent to the circle if it touches it just once. Such a line is called a tangent or a tangent line. The point where the tangent touches the circle is called the point of tangency.
• We can also speak of tangent segments or rays.
• One crucial property of tangents is that the radius drawn to the point of tangency is perpendicular to the tangent.
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Two Tangents
• Given a point P outside of a circle, there are two lines passing through P that are tangent to the circle.
• Also, the distance from P to each point of tangency is the same:
P
A B
PA PB
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Example
• From a point P outside of circle Q, draw a tangent to the point of tangency A.
• If then what is the radius of circle Q?
• Draw radius Note that is a right angle. So, by the Pythagorean Theorem:
P
A
Q
12 and 15,PA PQ
.AQPAQ
2 215 12 225 144 8 91AQ
12 15
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Example
• In the figure, is tangent to circle Q.
• If and then what is the radius of circle Q?
• Let r denote the radius. Then:
P
T
Q RS
PT
6PT 3,PS 6
3 r r
26 3(3 2 )r 36 9 6r 27 6r
27 / 6 r
So, radius 27 / 6 9 / 2 4.5
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Arcs
• An arc is an unbroken part of a circle.
• For example, in the figure, the part of the circle shaded red is an arc.
• A semicircle is an arc equal to half a circle.
• A minor arc is smaller than a semicircle.
• A major arc is larger than a semicircle.
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Naming Arcs
• A minor arc, like the one in red in the figure, can be named by drawing an arc symbol over its endpoints:
• Sometimes, to avoid confusion, a third point between the endpoints is used to name the arc:
• The reason for this is to avoid ambiguity because, given two points on a circle, there are two arcs between them (the long way around the circle or the short way).
A
B
P or .AB BA
or .APB BPA
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• Semicircles and major arcs must be named with three (or sometimes more) points.
• The arc highlighted in red in the figure would be called
It appears to be a major arc.
• If we wrote then we would be referring to the part of the circle that is not highlighted in red (a minor arc, it seems).
A
B
C
or .ABC CBA
AC
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The Measure of an Arc
• Each arc has a degree measure between 0 degrees and 360 degrees.
• A full circle is 360 degrees, a semicircle is 180 degrees, a minor arc measures less than 180 degrees, and a major arc measures more than 180 degrees.
• If an arc is a certain fraction of a circle, then its measure is the same fraction of 360 degrees. Some sample arc measures are given below.
C
A
E
B
DF
G
H
45mAB 90mABC 135mABD 180mACE 225mADF 270mADG
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Example
• In the figure, and
• Find
• Let denote the measure of each of the two equal arcs.
• Then
P
Q R
mPQ mPR 100 .mQR
.mPQ
100
x x
100 360x x
x
2 100 360x 2 260x
130mPQ x
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Central Angles
• Given a circle, a central angle is an angle whose vertex is at the center of the circle.
• In the figure, the center of the circle is and is a central angle that intercepts arc
• The measure of a central angle is equal to the measure of the arc it intercepts.
P
AQ
B
P APB
.AQB
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Inscribed Angles
• In a circle, an inscribed angle is an angle whose vertex is on the circle and whose sides are chords.
• In the figure, is an inscribed angle and it intercepts arc
• The measure of an inscribed angle is half the measure of its intercepted arc.
P
A
B
QAPB
.AQB
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Angle Inscribed in a Semicircle
• An important example of an inscribed angle is one that intercepts a semicircle.
• In the figure, is a diameter of the circle which divides the circle into two semicircles.
• intercepts a semicircle and since a semicircle measures the measure of the inscribed angle is
AB
APB
18090 .
A B
P
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A Special Kind of Inscribed Angle
• In the figure, is tangent to the circle.
• In this case, is called an inscribed angle and it intercepts arc
• And,
BT
ATB
T
A
B
PAPT
1.
2m ATB mAPT
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Example
• In the figure, P is the center of the circle and
• Find and
A
PB
C
D
30 .m A
, , , m B m CPD m BCD .m PCD
30
60
6060
Since inscribed intercepts ,
2 60 .
A CD
mCD m A
Then, since intercepts the
same arc, 0.5 60 30 .
B
m B
Then, since is a central angle
intercepting CD, 60 .
CPD
m CPD mCD
Since is inscribed in a semicircle,
90 .
BCD
m BCD
Since is isosceles with vertex ,
0.5(180 60 ) 60 .
CPD P
m PCD
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Example
• In the figure, O is the center of the circle,
and
• Find O
A
BC
D
P
55m ABC 2 .mAD mBC
.m DAB55
110
70
140
40
20
Since inscribed intercepts
, 2 55 110 .
ABC
AC mAC
Since is a semicircle,
180 110 70
ACB
mCB
Then 2 70 140 .mAD
Since is a semicircle,
180 140 40 .
ADB
mDB Since is an inscribed angle that
intercepts , 0.5(40 ) 20 .
DAB
DB m DAB
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Angles with Vertex Outside a Circle
• Given a circle, suppose point P is outside the circle and two secants are drawn from P as in the figure.
• Then intercepts two arcs:
• This angle and these arcs are related by the formula:
P
AB
CD
P and .AB CD
1
2m P mAB mCD
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More Angles with Vertex Outside a Circle
• The formula on the previous slide holds even if one or both sides of the angle are tangents instead of secants.
• In the figure, are tangents. Then:
and PS PT
P
S T
A
B
1 and
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2
m APS mAS mBS
m SPT mSAT mSBT
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Angles with Vertex Inside a Circle
• Consider in the figure. This angle intercepts Also, if you extend the sides of backwards then they intercept
• This angle and these arcs are related by the formula:
A
B C
D
P
APB
.ABAPB
.CD
1
2m APB mAB mCD
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Example
• In the figure, and
• Find
• Let denote the measures of respectively.
• Note A
B C
D
P
2mAD mBC84 .m APB
.mBC
and 2x x and ,BC AD
x
2x
8496
180 84 96 .m BPC 96 0.5( 2 ).x x 96 1.5x 96 /1.5 64mBC x