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
Copyright ? by Holt, Rinehart and Winston. All rights reserved.
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
Copyright ? by Holt, Rinehart and Winston. All rights reserved.
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
Copyright ? by Holt, Rinehart and Winston. All rights reserved.
45
Holt Geometry
Name
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
Copyright ? by Holt, Rinehart and Winston. All rights reserved.
46
Holt Geometry
Name
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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