Tangents to Circles ( with Circle Review)

Tangents to Circles(with Circle Review)

How do I identify segments and lines related to circles?

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

Essential Questions

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

Definitions

Diagram of Important Terms

diameter

radiusP

center

name of circle: P

Chord – a segment whose endpoints are points on the circle.

Definition

AB is a chord

B

A

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

Definition

MN is a secant

N

M

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

Definition

ST is a tangent

S

T

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

Example 1

FCB

G

A

H

D

E

Id. CE

c. DF

b. EI

a. AH tangent

diameter

chord

radius

Tangent circles – coplanar circles that intersect in one point

Definition

Concentric circles – coplanar circles that have the same center.

Definition

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

joins the centers of the two circles◦ Common external tangent – does not intersect the

segment that joins the centers of the two circles

Definitions

common external tangentcommon internal tangent

Tell whether the common tangents are internal or external.

Example 2

a. b.

common internal tangents common external tangents

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

More definitions

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

Definition

point of tangency

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

Perpendicular Tangent Theorem

l

Q

P

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

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.

Perpendicular Tangent Converse

l

Q

P

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

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

Definition

central angle

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

Definitions

Diagram of Arcs

CD B

Aminor arc: AB

major arc: ABD

semicircle: BAD

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.

Definitions

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

Arc Addition Postulate

A

C

B

mABC = mAB + mBC

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

Definition

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

Arcs and Chords Theorem

A

B

CAB BC if and only if AB BC

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

Perpendicular Diameter Theorem

D

F

GE

DE EF, DG FG

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

Perpendicular Diameter Converse

L

MJ

K

JK is a diameter of the circle.

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

Congruent Chords Theorem

G

F

E

C

D

B

A

AB CD if and only if EF EG.

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.

If two segments from the same exterior point are tangent to a circle, then they are congruent.

Congruent Tangent Segments Theorem

SP

R

T

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

Example 4

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

Find the measure of each arc.

Example 1

70PN L

M

a. LM

c. LMN

b. MNL

70°

360° - 70° = 290°

180°

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

Example 2

41

41

AC

D

E

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

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

Example 3

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

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

Definitions

inscribed angle

intercepted arc

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

Measure of an Inscribed Angle Theorem

C

A

D BmADB =

12

mAB

Find the measure of the blue arc or angle.

Example 1

RS

QT

a.

mQTS = 2(90 ) = 180

b.80

E

F G

mEFG = 12

(80 ) = 40

If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

Congruent Inscribed Angles Theorem

A

CB

DC 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

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

Circumscribed circle – A circle with an inscribed polygon.

Definitions

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

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.

Inscribed Right Triangle Theorem

A

C

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

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

Inscribed Quadrilateral Theorem

C

E

F

D G

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

Find the value of each variable.

Example 3

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 = 180y = 60

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.

Tangent-Chord Theorem

2 1

B

A

Cm1 = 12

mAB

m2 = 12

mBCA

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 =

12

(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

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.

Interior Intersection Theorem

m1 = 12

(mCD + mAB)

m2 = 12

(mAD + mBC)2

1

A

C

D

B

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.

Exterior Intersection Theorem

Diagrams for Exterior Intersection Theorem

1

BA

C

m1 = 12

(mBC - mAC)

2

P

RQ

m2 = 12

(mPQR - mPR)

3

XW

YZ

m3 = 12

(mXY - mWZ)

Find the value of x.

Example 3

174

106

x

P

R

Q

S

x = 12

(mPS + mRQ)

x = 12

(106+174 )

x = 12

(280)

x = 140

Find the value of x.

Try This!

120

40

x

T

R

S

U

x = 12

(mST + mRU)

x = 12

(40+120 )

x = 12

(160)

x = 80

Find the value of x.

Example 4

200

x 72

72 = 12

(200 - x )

144 = 200 - x

x = 56

Find the value of x.

Example 5

mABC = 360 - 92

mABC = 268 x92

C

AB

x = 12

(268 - 92)

x = 12

(176)

x = 88

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.

Chord Product Theorem

E

C

D

A

B

EA EB = EC ED

Find the value of x.

Example 1

x

96

3

E

B

D

A

C3(6) = 9x

18 = 9x

x = 2

Find the value of x.

Try This!

x 9

18

12E

B

D

A

C

9(12) = 18x

108 = 18x

x = 6

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.

Secant-Secant Theorem

C

A

B

ED

EA EB = EC ED

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.

Secant-Tangent Theorem

C

A

E

D

(EA)2 = EC ED

Find the value of x.

Example 2

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

Find the value of x.

Try This!

x

1012

11

H

GF

E

D

DE DF = DG DH

11(21) = 12(12 + x)

231 = 144 + 12x

87 = 12x

x = 7.25

Find the value of x.

Example 3

x

12

24

D

BC

A

CB2 = CD(CA)

242 = 12(12 + x)

576 = 144 + 12x

432 = 12x

x = 36

Find the value of x.

Try This!

3x5

10

Y

W

X Z

WX2 = XY(XZ)

102 = 5(5 + 3x)

100 = 25 + 15x

75 = 15x

x = 5