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Page 1: Circle  Segments and Volume

Circle Segments

and Volume

Page 2: Circle  Segments and Volume

Chords of Circles Theorem 1

Page 3: Circle  Segments and Volume

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

Page 4: Circle  Segments and Volume

Chord Arcs ConjectureIn the same circle, two minor arcs are congruent if and only if their corresponding chords are congruent.

CB

A

   IFF

 

IFF

andG

and

   

Page 5: Circle  Segments and Volume

8x – 7 3x + 3

8x – 7 = 3x + 3

Solve for x.

x = 2

Page 6: Circle  Segments and Volume

Find WX.4 2 3y y

4 3y

7y11WX cm

Example

Page 7: Circle  Segments and Volume

Find mAB

130º

Example

Page 8: Circle  Segments and Volume

Chords of Circles Theorem 2

Page 9: Circle  Segments and Volume

If a diameter is perpendicular to a chord, then it also bisects the chord and its arc.

This results in congruent arcs too.

Sometimes, this creates a right triangle & you’ll use Pythagorean Theorem.

Page 10: Circle  Segments and Volume

Perpendicular Bisector of a Chord Conjecture

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

D

G

FH

 

 

Page 11: Circle  Segments and Volume

IN Q, KL LZ. If CK = 2x + 3 and CZ = 4x, find x.

K

Q

C

L

Z

x = 1.52 3 4x x

Page 12: Circle  Segments and Volume

In P, if PM AT, PT = 10, and PM = 8, find AT.

T

AM

P

MT = 6AT = 12

22 28 10MT

Page 13: Circle  Segments and Volume

Chords of Circles Theorem 3

Page 14: Circle  Segments and Volume

Perpendicular Bisector to a Chord Conjecture

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

JK is a diameter of the circle.

J

L

K

M

Page 15: Circle  Segments and Volume

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

first chord is a diameter.

E

D

G

FDG

GF

, DE EF

Page 16: Circle  Segments and Volume

Chords of Circles Theorem 4

Page 17: Circle  Segments and Volume

In the same circle or in congruent circles two chords are congruent when they are equidistant from the center.

Page 18: Circle  Segments and Volume

•  

Chord Distance to the Center Conjecture

F

G

E

B

A

C

D

Page 19: Circle  Segments and Volume

In K, K is the midpoint of RE. If TY = -3x + 56 and US = 4x, find the length of TY.

Y

T

S

Kx = 8

U

RE

3 56 4x x 56 7x

TY = 32

Page 20: Circle  Segments and Volume

Example

CE =302 2 220 25x

15x

Page 21: Circle  Segments and Volume

Example

LN = 962 2 214 50x

48x

Page 22: Circle  Segments and Volume

Segment Lengths

in Circles

Page 23: Circle  Segments and Volume
Page 24: Circle  Segments and Volume

partpart

partpart

part part = part part Go down the chord and multiply

Page 25: Circle  Segments and Volume

9

2

6x

x = 3

Solve for x.

9 2 6x 18 6x

Page 26: Circle  Segments and Volume

Find the length of DB.

8

122x

3x x = 4

A

B

C

D

12 8 3 2x x 296 6x

216 x

DB = 20

Page 27: Circle  Segments and Volume

Find the length of AC and DB.

x = 8

x5

x – 4

10

A

B

C

D 5 10 4x x 5 10 40x x

5 40x

AC = 13

DB = 14

Page 28: Circle  Segments and Volume

outside whole = outside whole

Page 29: Circle  Segments and Volume

EA

B

C

D

7 13

4

x

7(7 + 13) 4 (4 + x)=

Ex: 3 Solve for x.

140 = 16 + 4x124 =

4xx = 31

Page 30: Circle  Segments and Volume

E

A

B

CD 8

5

6

x

6 (6 + 8)

5(5 + x)=

Ex: 4 Solve for x.

84 = 25 + 5x59 = 5x x =

11.8

Page 31: Circle  Segments and Volume

E

A

B

CD 4

x

8

10

x (x + 10)

8(8 + 4)=

Ex: 5 Solve for x.

x2+10x = 96x2 +10x – 96 =

0x = 6

Page 32: Circle  Segments and Volume

2tan = outside whole

Page 33: Circle  Segments and Volume

24

12 x

242 = 12 (12 + x)576 = 144 + 12x x = 36

Ex: 5 Solve for x.

Page 34: Circle  Segments and Volume

155

x

x2 = 5 (5 + 15)x2 = 100x = 10

Ex: 6


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