10.6 Segment Relationships in Circles - Big Ideas Learning
10.6 Segment Relationships in Circles
Essential Question
What relationships exist among the
segments formed by two intersecting chords or among segments of two
secants that intersect outside a circle?
Segments Formed by Two Intersecting Chords
Work with a partner. Use dynamic geometry software.
¡ª and DE
¡ª
a. Construct two chords BC
that intersect in the interior of a
circle at a point F.
Sample
REASONING
ABSTRACTLY
To be proficient in math,
you need to make sense
of quantities and their
relationships in problem
situations.
E
B
b. Find the segment lengths BF, CF,
DF, and EF and complete the table.
What do you observe?
F
?
BF
CF
BF CF
DF
EF
DF EF
A
C
D
?
c. Repeat parts (a) and (b) several times. Write a conjecture about your results.
Secants Intersecting Outside a Circle
Work with a partner. Use dynamic geometry software.
??
a. Construct two secants ??
BC and BD
that intersect at a point B outside
a circle, as shown.
Sample
C
b. Find the segment lengths BE, BC,
BF, and BD, and complete the table.
What do you observe?
A
?
BE
BC
BE BC
BF
BD
BF BD
E
B
F
D
?
c. Repeat parts (a) and (b) several times. Write a conjecture about your results.
Communicate Your Answer
D
E
9
A
18
3. What relationships exist among the segments formed by two intersecting chords
or among segments of two secants that intersect outside a circle?
F
4. Find the segment length AF in the figure at the left.
8
C
Section 10.6
int_math2_pe_1006.indd 613
Segment Relationships in Circles
613
1/30/15 11:58 AM
10.6 Lesson
What You Will Learn
Use segments of chords, tangents, and secants.
Core Vocabul
Vocabulary
larry
segments of a chord, p. 614
tangent segment, p. 615
secant segment, p. 615
external segment, p. 615
Using Segments of Chords, Tangents, and Secants
When two chords intersect in the interior of a circle, each chord is divided into two
segments that are called segments of the chord.
Theorem
Segments of Chords Theorem
If two chords intersect in the interior of a circle, then
the product of the lengths of the segments of one chord
is equal to the product of the lengths of the segments of
the other chord.
C
A
E
D
B
EA ? EB = EC ? ED
Proof Ex. 19, p. 618
Using Segments of Chords
Find ML and JK.
M
x+2
N
K x
?
?
NK NJ = NL NM
?
x (x + 4) = (x + 1) (x + 2)
x2
J
L
SOLUTION
?
x+4
x+1
+ 4x =
x2
+ 3x + 2
4x = 3x + 2
x=2
Segments of Chords Theorem
Substitute.
Simplify.
Subtract x2 from each side.
Subtract 3x from each side.
Find ML and JK by substitution.
ML = (x + 2) + (x + 1)
JK = x + (x + 4)
=2+2+2+1
=2+2+4
=7
=8
So, ML = 7 and JK = 8.
Monitoring Progress
Help in English and Spanish at
Find the value of x.
1.
2.
x 6
4 3
614
Chapter 10
int_math2_pe_1006.indd 614
2
4
x+1
3
Circles
1/30/15 11:58 AM
Core Concept
Tangent Segment and Secant Segment
R
external segment Q
secant segment
P
A tangent segment is a segment that
is tangent to a circle at an endpoint.
A secant segment is a segment that
contains a chord of a circle and has
exactly one endpoint outside the
circle. The part of a secant segment
that is outside the circle is called an
external segment.
tangent segment
S
PS is a tangent segment.
PR is a secant segment.
PQ is the external segment of PR.
Theorem
Segments of Secants Theorem
If two secant segments share the same endpoint
outside a circle, then 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.
A
B
E
C
D
EA ? EB = EC ? ED
Proof Ex. 20, p. 618
Using Segments of Secants
Find the value of x.
R
P
9
10
Q
11
x
S
T
SOLUTION
?
?
9 ? (11 + 9) = 10 ? (x + 10)
RP RQ = RS RT
180 = 10x + 100
80 = 10x
8=x
Segments of Secants Theorem
Substitute.
Simplify.
Subtract 100 from each side.
Divide each side by 10.
The value of x is 8.
Monitoring Progress
Help in English and Spanish at
Find the value of x.
3.
4.
9
x
x+1
5
Section 10.6
int_math2_pe_1006.indd 615
3
x+2
6
x?1
Segment Relationships in Circles
615
1/30/15 11:58 AM
Theorem
Segments of Secants and Tangents Theorem
A
If a secant segment and a tangent segment share an
endpoint outside a circle, then the product of the lengths
of the secant segment and its external segment equals
the square of the length of the tangent segment.
E
C
Proof Exs. 21 and 22, p. 618
D
?
EA2 = EC ED
Using Segments of Secants and Tangents
Find RS.
SOLUTION
ANOTHER WAY
In Example 3, you can draw
¡ª and QT
¡ª.
segments QS
R
?
162 = x (x + 8)
Q
16
?
x
x
Segments of Secants
and Tangents Theorem
RQ2 = RS RT
Q
16
R
8
S
T
Substitute.
256 =
x2
+ 8x
0=
x2
+ 8x ? 256
Simplify.
Write in standard form.
¡ª¡ª
S
?8 ¡À ¡Ì 82 ? 4(1)(?256)
x = ¡ª¡ª¡ª
Use Quadratic Formula.
2(1)
8
T
Because ¡ÏRQS and ¡ÏRTQ
intercept the same arc,
they are congruent. By
the Reflexive Property
of Congruence,
¡ÏQRS ? ¡ÏTRQ. So,
¡÷RSQ ¡« ¡÷RQT by the AA
Similarity Theorem. You can
use this fact to write and
solve a proportion to find x.
¡ª
x = ?4 ¡À 4¡Ì 17
Simplify.
Use the positive solution because lengths cannot be negative.
¡ª
So, x = ?4 + 4¡Ì17 ¡Ö 12.49, and RS ¡Ö 12.49.
Finding the Radius of a Circle
Find the radius of the aquarium tank.
B
20 ft
SOLUTION
?
CB2 = CE CD
Segments of Secants
and Tangents Theorem
?
202 = 8 (2r + 8)
Substitute.
400 = 16r + 64
Simplify.
336 = 16r
Subtract 64 from each side.
21 = r
r
D
r
E
8 ft
C
Divide each side by 16.
So, the radius of the tank is 21 feet.
Monitoring Progress
Help in English and Spanish at
Find the value of x.
5.
3
1
x
6.
5
7
x
x
7.
10
12
8. WHAT IF? In Example 4, CB = 35 feet and CE = 14 feet. Find the radius of
the tank.
616
Chapter 10
int_math2_pe_1006.indd 616
Circles
1/30/15 11:58 AM
10.6 Exercises
Dynamic Solutions available at
Vocabulary and Core Concept Check
1. VOCABULARY The part of the secant segment that is outside the circle is called a(n) _____________.
2. WRITING Explain the difference between a tangent segment and a secant segment.
Monitoring Progress and Modeling with Mathematics
In Exercises 3¨C6, find the value of x. (See Example 1.)
3.
15. ERROR ANALYSIS Describe and correct the error in
finding CD.
4.
?
x?3
12
10
10
6
x
18
9
5.
F
4
6.
8
x
2x
6
D
5
B
C
?
?
? ?
CD ? 4 = 15
CD DF = AB AF
15
x+8
3 A
CD 4 = 5 3
12
x+3
CD = 3.75
In Exercises 7¨C10, find the value of x. (See Example 2.)
7.
8.
16. MODELING WITH MATHEMATICS The Cassini
5
10
x
4
8
9.
10.
45
5
4
x+4
x?2
spacecraft is on a mission in orbit around Saturn until
September 2017. Three of Saturn¡¯s moons, Tethys,
Calypso, and Telesto, have nearly circular orbits of
radius 295,000 kilometers. The diagram shows the
positions of the moons and the spacecraft on one
of Cassini¡¯s missions. Find the distance DB from
¡ª is tangent to the circular
Cassini to Tethys when AD
orbit. (See Example 4.)
7
x
6
x
27
50
In Exercises 11¨C14, find the value of x. (See Example 3.)
11.
12.
x
7
13.
12
Calypso
C
B
D
Cassini
203,000 km
Saturn
x
12
A
T
l t
Telesto
14.
x+4
3
2
x
Section 10.6
int_math2_pe_1006.indd 617
Tethys
24
9
x
83,000 km
Segment Relationships in Circles
617
1/30/15 11:58 AM
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