Practice A 11-6 Segment Relationships in Circles - MR. BARNETT'S MATH ...

Name

Date

Class

LESSON Practice A 11-6 Segment Relationships in Circles

In Exercises 1?3, match the letter of the drawing to the formula that relates the

lengths of the segments in the drawing.

1. AC 2 AB (AD )

C

$

A.

"

!

#

%

2. AE(BE ) CE(DE )

B

3. AB(AD) AC(AE )

A

$

B.

" %

!#

C.

"

$

!

#

Find the value of the variable and the length of each chord.

4. *

-

X

.

+

5.

3

4

2

Y

,

x 6; JL 16; KM 17

0

1

y 25; PR 29; QS 20

6. Henri is riding a carousel at an amusement park. Devon, Emile, Francis, and George are looking on from around the edge of the carousel. At the moment shown in the figure, Devon is 2.5 meters from Henri, Emile is 1 meter from Henri, and Francis is 3.5 meters from Henri. Find the distance from Emile to George.

&

% $(

'

9.75 m

Find the value of the variable and the length of each secant segment.

7. 4

5 6

A

7

8

a 3.8; VT 11; VX 8.8

8.

&

Z

%

" #

$

z 4; BD 6; BF 6

Find the value of the variable.

9. (

X

) *

+

12

2

10.

3

4

Y

5

6

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43

Holt Geometry

Name

Date

Class

LESSON Practice B 11-6 Segment Relationships in Circles

Find the value of the variable and the length of each chord.

1. !

"X#

%

$

x 1; AD 6; BE 9

)

2.

Y

* (

& '

y 7; FH 8.3; GI 9.4

3. 0

4 3

1

Z

2

6

4.

7

5

M

9

z 7; PS 9.4; TR 9.4

8

m 4.5; UW 8.5; VX 9

Find the value of the variable and the length of each secant segment.

5. "

#

$

X%

&

x 4.5; BD 9.5; FD 9.5

6.

' )

(

Y

+

*

y 11.5; GJ 21; GK 17.5

7. 3

2

1

4

Z

5

z 19; SQ 18; SU 28

8.

%

N$

&

' #

n 8.25; CE 20.25; CF 27

Find the value of the variable. Give answers in simplest radical form if necessary.

+

9.

X

*

) (

1.5

10. +

,

.

Y

-

11. 52

26 Z

78

12.

9

5

B

10 70

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44

Holt Geometry

Name

Date

Class

LESSON Practice C 11-6 Segment Relationships in Circles

Find the value of x. Round to the nearest tenth if necessary.

1.

$

qi

2.

X$

&

#

#

X

%

"

2

'

5

3.

,

X

-

4.

7 5

4

X

6

.

13.3

3

7.2

5.

6.

!

"

#

X

$

%

7.1

'

)

*

(X

&

1.2

7.

+

X

,

0

-

.

8.4

8.

11

4

3

9

X

2

5

30?

6

15.7

Find each length.

9.

!

& "

AC

%

)

(

'

#

16.5

$

BD

16.8

10.

0

2 5

7

3

6

1

9

PY

40.5

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45

Holt Geometry

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Date

Class

LESSON Reteach 11-6 Segment Relationships in Circles

Chord-Chord Product Theorem

If two chords intersect in the interior of a

circle, then the products of the lengths of the

!

segments of the chords are equal.

$ %

"

#

AE EB CE ED

Find the value of x and the length of each chord.

HL LJ KL LM Chord-Chord Product Thm.

4 9 6 x

HL 4, LJ 9, KL 6, LM x

36 6x

Simplify.

6 x

Divide each side by 6.

HJ 4 9 = 13

KM 6 x

6 6 12

+

*

(

,

X

-

Find the value of the variable and the length of each chord.

1.

2

4

Y

5 3

6

2.

$

' X

% &

(

y 7; RS 10; TV 10

x 10; DF 16; GH 17

3.

-

*

Z %

.

,

z 7.5; JL 11; MN 11.5

4.

!

$

"X #

%

x 2.5; AC 18.5; DE 13

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46

Holt Geometry

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Date

Class

LESSON Reteach 11-6 Segment Relationships in Circles continued

? A secant segment is a segment of a _

secant with at least one endpoint on AE is a secant

the circle.

segment.

_

BE is an external

secant segment.

? An external secant segment is the part of the secant segment that lies in the exterior of the circle.

? A tangent segment is a segment of a tangent with one endpoint on the circle.

"

_

ED is a tangent segment.

$

If two segments intersect outside a circle, the following theorems are true.

Secant-Secant Product Theorem

The product of the lengths of one secant segment and

its external segment equals the product of the lengths

!

of the other secant segment and its external segment.

whole outside whole outside

AE BE CE DE

#

Secant-Tangent Product Theorem

The product of the lengths of the secant segment and

its external segment equals the length of the tangent

!

segment squared.

whole outside tangent2 AE BE DE 2

"

% $

"

%

$

Find the value of the variable and the length of each secant segment.

5.

6.

.

40

8 1

6

2X 3

x 2; NQ 12; NS 8

4 7

Z

5

8

6

9

8

9

z 12.25; TV 20.25; WV 18

Find the value of the variable. 7. &

12

'

4

(

X

*

8

Copyright ? by Holt, Rinehart and Winston. All rights reserved.

8.

.

9

+

6,

47

Y

-

7.5

Holt Geometry

Name

Date

Class

LESSON Reteach 11-1 Lines That Intersect Circles

Lines and Segments That Intersect Circles

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

_

_

AB and CD

"

? A secant is a line that intersects a circle are chords.

at two points.

!

? A tangent is a line in the same plane as a circle that intersects the circle at exactly one point, called the point of tangency.

# %

? Radii and diameters also intersect circles.

CD is a secant.

$

is a tangent.

E is a point of tangency.

Tangent Circles

Two coplanar circles that intersect at exactly one point are called tangent circles.

points of tangency

Identify each line or segment that intersects each circle.

1.

'

2.

+

.

(

&

_ M chord: F_G; secant_: ; tang_ent: m;

diam.: FG ; radii: HF and HG

* -

,

_

chord:

LM

;

s_ecant:

LM

;

tangent:

MN ; radius: JK

Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at that point.

3.

Y

4.

Y

2

.

X

2 0

2

3

34

X

0

2

0

3

N : r 3; P: r 1; pt. of

tangency: (1, 2); tangent line:

y 2

Copyright ? by Holt, Rinehart and Winston. All rights reserved.

6

S: r 4; T: r 2; pt. of tangency: (7, 0); tangent line: x 7

Holt Geometry

Name

Date

LESSON Reteach 11-1 Lines That Intersect Circles continued

Theorem

If two segments are tangent to a circle from the same external point, then the segments are congruent.

Hypothesis

&

%

#

'

_

_

EF and EG are tangent to C.

In the figure above, EF 2y and EG y 8. Find EF.

EF EG

2 segs. tangent to from same ext. pt. segs. .

2y y 8 Substitute 2y for EF and y 8 for EG.

y 8

Subtract y from each side.

EF 2(8)

EF 2y; substitute 8 for y.

16

Simplify.

Class

Conclusion

_ _

EF EG

The segments in each figure are tangent to the circle. Find each length.

5. BC

6. LM

" X

Y -

!

#

X $

Y ,

6

. +

14

7. RS

3

Y

0

2

Y 4

10

Copyright ? by Holt, Rinehart and Winston. All rights reserved.

8. JK

*

X '

+

(

X

19

7

Holt Geometry

Name

Date

Class

LESSON Reteach 11-2 Arcs and Chords

Arcs and Their Measure

? A central angle is an angle whose vertex is the center of a circle. ? An arc is an unbroken part of a circle consisting of two points on a circle and all the points

on the circle between them.

ADC is a major arc. m ADC 360? mABC

360? 93? 267?

$ " #

ABC is a central angle.

AC is a minor arc m AC mABC 93?.

? If the endpoints of an arc lie on a diameter, the arc is a semicircle and its measure is 180?.

Arc Addition Postulate

The measure of an arc formed by two adjacent arcs

!

"

is the sum of the measures of the two arcs.

m ABC m AB m BC

#

Find each measure.

(

'

*

+ &

1. m HJ 2. m FGH

63? 117?

5. m LMN 6. m LNP

75? 225?

Copyright ? by Holt, Rinehart and Winston. All rights reserved.

#

"

$

&

!

%

3. m CDE 4. m BCD

130? 140?

.

-

2

0

1 ,

14

Holt Geometry

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