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10.6 Segment Lengths in Circles
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Segments of a chord: The segments resulting when two
chords intersect inside a circle.
C
E
D
B
A
F
Segments of a chord Rule
(Theorem)
The product of the segments of one
chord is equal to the product of the
segments of the other chord.
BC times CD = AC times CF
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GUIDED PRACTICE
Find the value(s) of x.
x (3) = (4) (6) Substitute.
Simplify.
x = 8 Solve for x.
SOLUTION
3x = 24
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for Examples 1 and 2 EXAMPLE 1
ALGEBRA Find ML and JK.
SOLUTION
NK NJ = NL NM
x (x + 4) = (x + 1) (x + 2)
x2 + 4x = x2 + 3x + 2
4x = 3x + 2
x = 2
ML = ( x + 2 ) + ( x + 1)
= 2 + 2 + 2 + 1
= 7
JK = x + ( x + 4)
= 2 + 2 + 4
= 8
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Secant segment: segment containing a chord of a circle
and has exactly one endpoint outside the circle.
B
C
A
D
E A secant segment has an external
segment and an internal segment.
Segments of secants theorem:
Outside times Whole = Outside times Whole
CD ∙ CE = CB ∙ CA
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GUIDED PRACTICE
Find the value(s) of x.
SOLUTION
Use Theorem 10.15. 5 (x + 5) = 6 (6 + 9)
90 = 5x + 25 Simplify.
13 = x Solve for x
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GUIDED PRACTICE
Find the value(s) of x.
SOLUTION
Use Theorem 10.15. 3 (3 + x + 2) = (x+1)(x + 1 + x – 1)
3x + 15 = 2x2 + 2x Simplify.
0 = 2x2 – x – 15 Combine Like Terms.
0 = (x – 3) (2x + 5) Solve for x
3 = x Use positive solution
CP
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The same rule applies for Secants and/or Tangents.---
Except the outside is the whole on a tangent segment.
E
H
G
F
D
FE ∙FD = FG ∙ FG
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GUIDED PRACTICE
Find the value of x.
SOLUTION
x2 = 1 (1 + 3)
x = 2
Use Theorem 10.16.
Simplify.
Simplify.
4.
= 4 x2
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GUIDED PRACTICE
Find the value of x.
SOLUTION
5.
49 = 25 + 5x
24 = 5x
72 = 5 (x + 5) Use Theorem 10.16.
Simplify.
Write in standard form.
Simplify. x = 24 5
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GUIDED PRACTICE
Determine which theorem you would use to find x. Then
find the value of x.
8.
Use Theorem 10.14.
x (18) = (9) (16) Substitute.
18x = 144 Simplify.
SOLUTION
Simplify. x = 8
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GUIDED PRACTICE
Find the value of x.
SOLUTION
6.
144 = x2 + 10x
0 = x2 + 10x – 144
122 = x (x + 10)
x –10 + 102 – 4(1) (– 144)
2(1) =
Use Theorem 10.16.
Simplify.
Write in standard form.
Use quadratic formula
Or factor.
Simplify. x = 8
CP
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EXAMPLE 4 Solve a real-world problem
SCIENCE
Tethys, Calypso, and Telesto are three of Saturn’s
moons. Each has a nearly circular orbit 295,000
kilometers in radius. The Cassini-Huygens spacecraft
entered Saturn’s orbit in July 2004. Telesto is on a
point of tangency. Find the distance DB from Cassini
to Tethys.
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EXAMPLE 4 Solve a real-world problem
SOLUTION
DC DB = AD2 Use Theorem 10.16.
83,000 DB 203,0002 Substitute.
DB 496,494 Solve for DB.
ANSWER
Cassini is about 496,494 kilometers from Tethys.
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Geometry
• Page 692 (2-20ev, 30,34-42ev)
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Sophomore Math
• worksheet