Circle segments and volume
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Circle Segments and Volume. Chords of Circles Theorem 1. In the same circle, or in congruent circles two minor arcs are congruent if and only if their corresponding chords are congruent. Chord Arcs Conjecture.

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Circle Segments and Volume


Chords of Circles Theorem 1


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


Chord Arcs Conjecture

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

IFF

G

and

IFF

and


Solve for x.

8x – 7

3x + 3

8x – 7 = 3x + 3

x = 2


Example

Find WX.


Example

Find

130º


Chords of Circles Theorem 2


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.


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

Perpendicular Bisector of a Chord Conjecture

H


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

x = 1.5

Q

C

Z

K

L


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

P

A

M

MT = 6

T

AT = 12


Chords of Circles Theorem 3


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

Perpendicular Bisector to a Chord Conjecture

  • is a diameter of the circle.


  • ,

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


Chords of Circles Theorem 4


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


Chord Distance to the Center Conjecture


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

U

T

K

E

x = 8

R

S

Y

TY = 32


Example

30

CE =


Example

LN =

96


Segment Lengths in Circles


Two chords intersect

INSIDE the circle

Type 1:

part

part

part

part

Go down the chord and multiply


Solve for x.

9

6

x

2

x = 3


12

2x

8

3x

Find the length of DB.

A

D

x = 4

C

DB = 20

B


Find the length of AC and DB.

D

A

x – 4

x

C

5

10

x = 8

AC = 13

DB = 14

B


Two secants intersect

OUTSIDE the circle

Type 2:


Ex: 3 Solve for x.

B

13

A

7

E

4

C

x

D

7

(7 + 13)

=

4

(4 + x)

x = 31

140 = 16 + 4x

124 = 4x


Ex: 4 Solve for x.

B

x

A

5

D

8

6

C

E

6

(6 + 8)

=

5

(5 + x)

84 = 25 + 5x

x = 11.8

59 = 5x


Ex: 5 Solve for x.

B

10

A

x

D

4

8

C

E

x

(x + 10)

=

(8 + 4)

8

x2+10x = 96

x = 6

x2 +10x – 96 = 0


Type 2 (with a twist):

Secant and Tangent


Ex: 5 Solve for x.

x

12

24

(12 + x)

242

=

12

x = 36

576 = 144 + 12x


Ex: 6

5

15

x

(5 + 15)

x2

=

5

x2 = 100

x = 10


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