Circles

Chapter 10

Essential Questions

How do I identify segments and lines related to circles?

How do I use properties of a tangent to a circle?

Definitions

A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle.

Radius – the distance from the center to a point on the circle

Congruent circles – circles that have the same radius.Diameter – the distance across the circle through its

center

Diagram of Important Terms

diameter

radiusP

center

name of circle: P

Definition

Chord – a segment whose endpoints are points on the

circle.

AB is a chord

B

A

Definition

Secant – a line that intersects a circle in two points.

MN is a secant

N

M

Definition

Tangent – a line in the plane of a circle that intersects the circle in exactly one point.

ST is a tangent

S

T

Example 1

Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius.

FC

B

G

A

H

D

E

Id. CE

c. DF

b. EI

a. AH tangent

diameter

chord

radius

Definition

Tangent circles – coplanar circles that intersect in one point

Definition

Concentric circles – coplanar circles that have the same center.

Definitions

Common tangent – a line or segment that is tangent to two coplanar circlesCommon internal tangent – intersects the segment

that joins the centers of the two circlesCommon external tangent – does not intersect the

segment that joins the centers of the two circles

common external tangentcommon internal tangent

Example 2

Tell whether the common tangents are internal or external.

a. b.

common internal tangents common external tangents

More definitions

Interior of a circle – consists of the points that are inside the circle

Exterior of a circle – consists of the points that are outside the circle

Definition

Point of tangency – the point at which a tangent line intersects the circle to which it is tangent

point of tangency

Perpendicular Tangent Theorem

If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

l

Q

P

If l is tangent to Q at P, then l QP.

Perpendicular Tangent Converse

In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

l

Q

P

If l QP at P, then l is tangent to Q.

Congruent Tangent Segments TheoremIf two segments from the same exterior

point are tangent to a circle, then they are congruent.

If SR and ST are tangent to P, then SR ST.

SP

R

T

Example

AB is tangent to C at B.AD is tangent to C at D.

Find the value of x. 11

x2 + 2

AC

D

BAD = AB

x2 + 2 = 11

x2 = 9

x = 3

Definition

Central angle – an angle whose vertex is the center of a circle.

central angle

Definitions

Minor arc – Part of a circle that measures less than 180°

Major arc – Part of a circle that measures between 180° and 360°.

Semicircle – An arc whose endpoints are the endpoints of a diameter of the circle.

Note : major arcs and semicircles are named with three points and minor arcs are named with two points

Diagram of Arcs

CD B

Aminor arc: AB

major arc: ABD

semicircle: BAD

Definitions

Measure of a minor arc – the measure of its central angle

Measure of a major arc – the difference between 360° and the measure of its associated minor arc.

Arc Addition Postulate

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

A

C

B

mABC = mAB + mBC

Definition

Congruent arcs – two arcs of the same circle or of congruent circles that have the same measure

SP

R

T

Arcs and Chords Theorem

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

A

B

C

AB BC if and only if AB BC

Perpendicular Diameter Theorem

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

D

F

G

EDE EF, DG FG

Perpendicular Diameter Converse

If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

L

M

J

K

JK is a diameter of the circle.

Congruent Chords Theorem

In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

G

F

E

C

D

B

A

AB CD if and only if EF EG.

Right Triangles Pythagorean Theorem

45

43

11

D

E

C

Radius is perpendicular to the tangent. < E is a right angle

Example 3

Tell whether CE is tangent to D.

45

43

11

D

E

C

Use the converse of the Pythagorean Theorem to see if the triangle is right.

112 + 432 ? 452

121 + 1849 ? 2025

1970 2025

CED is not right, so CE is not tangent to D.

Example 1

Find the measure of each arc.

70PN L

M

a. LM

c. LMN

b. MNL

70°

360° - 70° = 290°

180°

Example 2

Find the measures of the red arcs. Are the arcs congruent?

41

41

AC

D

EmAC = mDE = 41Since the arcs are in the same circle, they are congruent!

Example 3

Find the measures of the red arcs. Are the arcs congruent?

81

C

A

D

E

mDE = mAC = 81However, since the arcs are not of the same circle orcongruent circles, they are NOT congruent!

Example 4

A

(2x + 48)(3x + 11)

D

B

C

3x + 11 = 2x + 48

Find mBC.

x = 37

mBC = 2(37) + 48

mBC = 122

Definitions

Inscribed angle – an angle whose vertex is on a circle and whose sides contain chords of the circle

Intercepted arc – the arc that lies in the interior of an inscribed angle and has endpoints on the angle

inscribed angle

intercepted arc

Measure of an Inscribed Angle Theorem

If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.

C

A

D BmADB =

1

2mAB

Example 1

Find the measure of the blue arc or angle.

RS

QT

a.

mQTS = 2(90 ) = 180

b.80

E

FG

mEFG = 1

2(80 ) = 40

Congruent Inscribed Angles TheoremIf two inscribed angles of a circle intercept

the same arc, then the angles are congruent.

A

CB

D

C D

Example 2

It is given that mE = 75 . What is mF?

D

E

HF

Since E and F both interceptthe same arc, we know that theangles must be congruent.

mF = 75

Definitions

Inscribed polygon – a polygon whose vertices all lie on a circle.

Circumscribed circle – A circle with an inscribed polygon.

The polygon is an inscribed polygon and the circle is a circumscribed circle.

Inscribed Right Triangle Theorem

If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.

A

C

BB is a right angle if and only if ACis a diameter of the circle.

Inscribed Quadrilateral Theorem

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

C

E

F

DG

D, E, F, and G lie on some circle, C if and only if mD + mF = 180 and mE + mG = 180 .

Example 3

Find the value of each variable.

2x

Q

A

B

C

a.

2x = 90

x = 45

b. z

y

80

120

D

E

F

G

mD + mF = 180

z + 80 = 180

z = 100

mG + mE = 180

y + 120 = 180

y = 60

Tangent-Chord Theorem

If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

21

B

A

Cm1 = 1

2mAB

m2 = 1

2mBCA

Example 1

m

102

T

R

S

Line m is tangent to the circle. Find mRST

mRST = 2(102 )

mRST = 204

Try This!

Line m is tangent to the circle. Find m1

m

150

1

T

Rm1 =

1

2(150 )

m1 = 75

Example 2

(9x+20)

5x

D

B

CA

BC is tangent to the circle. Find mCBD.

2(5x) = 9x + 20

10x = 9x + 20

x = 20

mCBD = 5(20 )

mCBD = 100

Interior Intersection Theorem

If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

m1 = 1

2(mCD + mAB)

m2 = 1

2(mAD + mBC)

21

A

C

D

B

Exterior Intersection Theorem

If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

Diagrams for Exterior Intersection Theorem

1

BA

C

m1 = 1

2(mBC - mAC)

2

P

RQ

m2 = 1

2(mPQR - mPR)

3

X

W

YZ

m3 = 1

2(mXY - mWZ)

Example 3

Find the value of x.

174

106

x

P

R

Q

S

x = 1

2(mPS + mRQ)

x = 1

2(106+174 )

x = 1

2(280)

x = 140

Try This!

Find the value of x.

120

40

x

T

R

S

U

x = 1

2(mST + mRU)

x = 1

2(40+120 )

x = 1

2(160)

x = 80

Example 4

Find the value of x.

200

x 72

72 = 1

2(200 - x )

144 = 200 - x

x = 56

Example 5

Find the value of x.

mABC = 360 - 92

mABC = 268 x92

C

AB

x = 1

2(268 - 92)

x = 1

2(176)

x = 88

Chord Product Theorem

If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

E

C

D

A

B

EA EB = EC ED

Example 1

Find the value of x.

x

96

3

E

B

D

A

C3(6) = 9x

18 = 9x

x = 2

Try This!

Find the value of x.

x 9

18

12

E

B

D

A

C

9(12) = 18x

108 = 18x

x = 6

Secant-Secant Theorem

If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment.

C

A

B

ED

EA EB = EC ED

Secant-Tangent Theorem

If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.

C

A

E

D

(EA)2 = EC ED

Example 2

Find the value of x.

LM LN = LO LP

9(20) = 10(10+x)

180 = 100 + 10x

80 = 10x

x = 8 x

10

11

9

O

M

N

L

P

Try This!

Find the value of x.

x

1012

11

H

GF

E

D

DE DF = DG DH

11(21) = 12(12 + x)

231 = 144 + 12x

87 = 12x

x = 7.25

Example 3

Find the value of x.

x

12

24

D

BC

A

CB2 = CD(CA)

242 = 12(12 + x)

576 = 144 + 12x

432 = 12x

x = 36

Try This!

Find the value of x.

3x5

10

Y

W

X Z

WX2 = XY(XZ)

102 = 5(5 + 3x)

100 = 25 + 15x

75 = 15x

x = 5