M—06/01/09—HW #75: Pg 654-655: 1-20, skip 9, 16. Pg 821-822: 16—38 even, 39—45 odd2) C 4) B 6) C 8) A 10) 5 12) EF = 40, FG = 50, EH = 24014) (3\2, 2); (5\2) 18) BC, DC 20) BH = 106, HD = 74

16) 35 18) 125 20) 90 22) 9024) 65 26) x = 80, y = 100 28) 9030) 40 32) 46 34) 3 36) 2538) 6

Chapter 10 Review

• Angles – Middle Same

• Inside – Add, div 2

• On – Half

• Outside, subtract, div 2

• Segments – Tan congruent

• Inside, part part = part part

• Outside, part whole = part whole pr

• Part whole = whole squared

PBPAThen

O o tangent tare PB,PA :Given

P

Theorem 10.3

Tangents to a circle from a point are congruent.

O

A

B

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

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

angle

whole

xy

zsup

100

angleoppquadinsc

90

90

semiangleinsc

90

90

semiangleinsc

Theorem 10-12

Measure of angle formed by chord and tangent is half the intercepted arc

Chord tan half arc

45

5.22

1

Thrm: Measure of angles formed by two chords that intersect inside the circle is equal to half the sum of the measures of the intercepted arcs.

A

B

C

D

)(2

11 mCDmABm

26

50

38

x

x

x

y

y

y

1

2

3

The measure of an angle formed by two secants, two tangents, or a secant and tangent drawn from a point OUTSIDE the circle is equal to half the difference of the measures of the intercepted arcs.

x y1m )(2

1

x y2m )(2

1

x y3m )(2

1

40

x

30

y

100

z

80, inscribed angle

25, .5(80-30)

50, .5(100-z)=25

80, .5(x+160)=120

40, .5(160-80)=y

120xy

160

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

ca

d

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

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

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

• M—06/01/09—HW #75: Pg 654-655: 1-20, skip 9, 16. Pg 821-822: 16—38 even, 39—45 odd

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