CIRCLES( Secant, Tangent and ETC.)

8/12/2019 CIRCLES( Secant, Tangent and ETC.)

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10.1Properties of Tangents

Circle

Secant

Tangent

Radius

Diameter

Chord

Radius

Chord

Diameter

Secant

Tangent

Example

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

A

D

C

E

B

Fa.

b.

c.

d.


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3/20

In K, J is a point of tangency. Find the radius of K.

36 cm

48 cm

r

rK

J L

Theorem 10.2

Find x.

6x - 8

25


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10.1 Homework

3-10 13.

19. 23.

24. 25.

26. 29.


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34.

37.


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10.2Find Arc Measures

Central angle

central angle

Minor arc

Major arc

major arc ADB

minor arc AB

B

A

C

D

Semicircle

65

G

E

F

Measuring Arcs

Measure of a major arc

Find the measure of each arc of K where HJ is a diameter.

80

H

K

J

Ia.b.c.d.


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Examplea result of a survey about the ages of people in a town are shown. Find the indicated arcmeasures.

17-44

15-17

45-64

>65

Ages of People (in years)

8060

10090

QU

TS

RV

a. mRUb. mRSTc. mRVTd. mUST

Congruent circles

Congruent arcs

Tell whether arcs CD and EF are congruent. Why?

45 45CP

F

EDa. b. c.

F

E

CP

D

110

D

C

Q

F

E


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10.2 HW PICTURES:

3 10

17. 20.

21.


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10.3Properties of Chords

Theorem 10.3

Theorem 10.4

Theorem 10.5

108

A

B

C

D

Examples

1. In R mAB = _______ . Find the mCD.

2. Use the diagram of C to 3. In the diagram of P, PV = PW,

find the length of BF . QR = 2x + 6, and ST = 3x 1. Find QR.

W

V

PQ

R

S

T

F15 B

G

C

A

D


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17. 21.


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23.

30.


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10.4Inscribed angles and Polygons

Inscribed angle

D

C

B

A

Intercepted arc

Theorem 10.7Measure of an Inscribed Angle Theorem

E

D

C

B

A

Theorem 10.8

Theorem 10.9

Q

T

S

R

--Conversely,


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Theorem 10.10

Examples

1. Find the indicated measures in X

104

33

X

U

V

W

Y

a. mUW b. mVWY

2. Find mWX and m WYX.

44A

Z

Y

X

W

Find the measure of each angle in the quadrilateral.

3.

(5y - 5)

(4y + 5)

5x

7x


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10.5Other Angle Relationships in Circles

D

2

1

B

C

A

Theorem 10.11

Find the indicated measures.

mAB = 124

D

2

1

B

C

A

Intersecting lines and circlesif two lines intersect a circle, there are three places where thelines can intersect

Theorem 10.12Angles Inside the Circle

2

1

E

G

D

F


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Theorem 10.13Angles Outside the Circle

Examples

Line m is tangent to the circle. Find x or y.

y

118

1. 2.

63

89x

(17x + 6)(7x - 2)39

228

x

3. 4.

x

44

30

5.


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1

E

D

C

B

A

0.6Finding Segment Lengths in Circles

ind x.

gment

egment

Theorem 10.14Segments of Chords

F

Secant se

External s

Theorem 10.15Segments of Secants Theorem


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F

x

21

x27R

T

V

ind x.

ind RT.

ind x.

F

10.16Segments of Secants and Tangents Theorem

F

x

x - 4

x

24


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1

-10 -5 5 10

6

4

2

-2

-4

-6

-8

0.7Write and Graph Equations of Circles

ircle Centered at the Origin:

tandard Equation of a Circle Centered at (h, k):

rite the equations of the circles using the given information.

1. Center (0, 0), radius 5

2. Center (-3, 8), radius 5/3

3. Center (1, 2) and a point on the circle is (4, 2)

Standard Equation of aC

S

3

2

1

-1

-4 -2 2 4

h k

-

2

W


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Graph each equation.

. x2+ y2= 25 5. (x - 2)2+ (y + 1)2= 44

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CIRCLES( Secant, Tangent and ETC.)