1

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

5

• 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

21

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