10.1 – Tangents to Circles

A circle is a set of points in a plane at a given distance from a given point in the plane.

The given point is a center.

CENTER

The given distance is the radius.

RADIUS

More than one is called the radii

All radii are congruent.

Two circles are congruent if the radius are congruent.

O

A chord is a segment whose endpoints lie on a circle.

A secant is a line that contains a chord.

SECANT

A diameter is a chord that contains the center of a circle.

DIAMETER

CHORDA tangent is a line in the plane of the circle that intersects the circle at EXACTLY one point, called the point of tangency. tangency.ofpoint theisA

tangents.called are PAsegment and PAray tangent The

P

A

A line tangent to each of two coplanar circles is called a common tangent.

Common internal tangent intersects the segment joining the centers.

Common external tangent DOES NOT intersect the segment joining the centers.

Segment joining the centers

Tangent circles are coplanar circles that are tangent to the same line at the same point

External tangency. Internal tangency.

Congruent circles and spheres have the same radii.

Concentric circles lie in the same plane and have the same center.

Concentric circles.

The green one is not concentric

Theorem 10-1

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

m

OT

m

m

OTThen

Tat O o tangent tis :Given

Theorem 10-2

If a line in a plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to a circle.

O o tangent tis Then

Tat OT radius

O, of planein line :Given

m

m

m

PBPAThen

O o tangent tare PB,PA :Given

P

Theorem 10.3

Tangents to a circle from a point are congruent.

O

A

B

P O

A

B

10.2 – Arcs and Chords

Central angle is an angle with vertex at center.

Arc is unbroken part of a circle.

Y

OZ

The smaller arc is called the MINOR ARC.

The larger arc is called the major arc, and generally uses 3 letters to name it.

W

YZ

YWZ

A

If A and Z are endpoints of a diameter, it is a semi-circle, since both arcs are the same, there is no major or minor

Y

OZ

W

The measure of the minor arc is DEFINED as the measure of its central angle.

The measure of the major arc is 360 – (Central angle)

Semicircles have a measure of 180o.

Adjacent arcs have exactly one point in common.

YZ ZW are Adjacent

arcs. two theof

measure theof sum theis arcsadjacent

by two formed arc theof measure The

PostulateAddition Arc

mYZ mZW mYZW + =Discuss notation

Congruent arcs are arcs in the same circle OR CONGRUENT circles that have equal measures.

CONGRUENT

NOT CONGRUENT

9090

In a same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent.

(Note, if and only if, it’s biconditional, so the converse is also true)

A

B

C

R

S

QQSACThen

QRSABCIf

A

B C

DE

F

______

___________

___________

||

40,45:

mED

mFDmFBD

mEFDmFD

Find

BCED

EAFmDBCmGiven

Also state if they are major or minor arcs

W

R

NG

O

Theorem 10.4

In the same circle or in congruent circles:

1) Congruent arcs have congruent chords.

2) Congruent chords have congruent arcs.

B

A

N

E

R

Theorem 10.5

A diameter that is perpendicular to a chord bisects the chord and its arc.

Theorem 10.6

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

B

A

N

E

R

Theorem 10.7

In the same circle or in congruent circles.

1) Chords equally distant from the center (or centers) are congruent.

2) Congruent chords are equally distant from the center (or centers)

Remember, shortest distance means perpendicular.

45

x

y

z

chords bisectsdiameter

4x

segmentscongruent

npythagorea Some

3y

congruent.

aret equidistan Chords

8z

10.3 – Inscribed Angles

Def: An inscribed angle is an angle whose vertex is on the circle and sides are chords in the circle.

Theorem: Measure of inscribed angle is half the measure of the intercepted arc.

halfinsc

45

5.22

Look at the pictures, what can you conclude?

12

1

12

3

4

Two inscribed angles intercept same arc, then they are congruent

21

Angle inscribed in semi circle is 90 degrees

Proof: 180\2

901m

If quadrilateral is inscribed in circle, opposite angles supplementary

supp3,2

supp4,1

arcsameinsc 90semiinsc supoppquadinsc

80

yz

arc

halfisangleinsc

40

congruentare

arcsameanginsc

40

80

angle

whole

xy

zsup

100

angleoppquadinsc

90

90

semiangleinsc

90

90

semiangleinsc

10.5 – Segment Lengths in Circles

10.15 Two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. a

bc

da

bc

d

a

105

30

150=10a

15=a

b

cad

Theorem 10.16: When two secants are drawn from an external point, the product of one secant with external segment is equal to the product of the other secant and external segment.

External Segment

Theorem 10.17: When a secant and tangent are drawn to a circle from an external point, the product of the secant and its external segment is equal to the tangent square.

b

a c

ab=cd

ab=c2

A

B

C

D

F

E

?

6?

1012

FE

BFBC

CDAB

8

18

6

816

ECxED

EBAD

4 or 12

A

B

C

D

E

124

)12)(4(0

48160

4816

68)16(

2

2

xorx

xx

xx

xx

xx

10.6 – Equations of Circles

Equation of a circle

(x – h)2 + (y – k)2 = r2

(h,k) is the center

r is the radius

Graphing a circle, give radius, center

(x – 3)2 + (y + 1)2 = 25

(h,k) is the center

r is the radius1)Plot center

2) Find radius

3) Plot points

4) Connect

(3,–1)

525

Graphing a circle, give radius, center

(x + 2)2 + (y – 3)2 = 10

(h,k) is the center

r is the radius1)Plot center

2) Find radius

3) Plot points

4) Connect

(–2,3)

16.310

Write equation given center and radius

(x – 2)2 + (y + 4)2 = 21

(2,–4)

21

1) Center

2) Radius

(x – 2)2 + (y –(– 4))2 =( )2

(x + 1)2 + (y – 2)2 = 9

(–1,2)

3

1) Center

2) Radius

(x – (– 1))2 + (y – 2)2 = 3221

Write equation of circle given picture

(x + 1)2 + (y – 2)2 = 9

(–1,2)

3

1) Center

2) Radius

(x – (– 1))2 + (y – 2)2 = 32

Write equation given center and point on circle.

1) Use distance formula to find radius

2) Then same as before.

Center (1,–3)

Point on Circle (–4 ,2)

22 )23())4(1( r22 )5()5( r

2525r

50r

222 )50())3(()1( yx

50)3()1( 22 yx

• In Circle, On circle, Outside Circle

• Locus – Circles, lines, and triangles, and basic shapes

Describe the set of points 2 cm from point A

Describe the set of points 1 cm from line l

Describe the set of points equidistant from points A and B

Describe the set of points equidistant to the sides and in the interior of the angle

All the points equidistant from points A, B, and C