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Segments of Chords, Secants,
and Tangents
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Printed: February 19, 2013
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C ONCEPT
Concept 1. Segments of Chords, Secants, and Tangents
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Segments of Chords,
Secants, and Tangents
Learning Objectives
• Find the lengths of segments associated with circles.
In this section we will discuss segments associated with circles and the angles formed by these segments. The figures
below give the names of segments associated with circles.
Segments of Chords
Theorem If two chords intersect inside the circle so that one is divided into segments of length a and b and the other
into segments of length b and c then the segments of the chords satisfy the following relationship: ab = cd.
This means that the product of the segments of one chord equals the product of segments of the second chord.
Proof
We connect points A and C and points D and B to make 4AEC and 4DEB.
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6
6
6
AEC ∼
= 6 DEB
∼ 6 BDC
CAB =
ACD ∼
= 6 ABD
Vertical angles
Inscribed angles intercepting the same arc
Inscribed angles intercepting the same arc
Therefore, 4AEC ∼ 4DEB by the AA similarity postulate.
In similar triangles the ratios of corresponding sides are equal.
c a
= ⇒ ab = cd
b d
Example 1
Find the value of the variable.
10x = 8 × 12
10x = 96
x = 9.6
Segments of Secants
Theorem If two secants are drawn from a common point outside a circle and the segments are labeled as below, then
the segments of the secants satisfy the following relationship:
a(a + b) = c(c + d)
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Concept 1. Segments of Chords, Secants, and Tangents
This means that the product of the outside segment of one secant and its whole length equals the product of the
outside segment of the other secant and its whole length.
Proof
We connect points A and D and points B and C to make 4BCN and 4ADN.
6
6
BNC ∼
= 6 DNA
NBC ∼
= 6 NDA
Same angle
Inscribed angles intercepting the same arc
Therefore, 4BCN ∼ 4DAN by the AA similarity postulate.
In similar triangles the ratios of corresponding sides are equal.
a c+d
=
⇒ a(a + b) = c(c + d)
c a+b
Example 2
Find the value of the variable.
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10(10 + x) = 9(9 + 20)
100 + 10x = 261
10x = 161
x = 16.1
Segments of Secants and Tangents
Theorem If a tangent and a secant are drawn from a point outside the circle then the segments of the secant and the
tangent satisfy the following relationship
a(a + b) = c2 .
This means that the product of the outside segment of the secant and its whole length equals the square of the tangent
segment.
Proof
We connect points C and A and points B and C to make 4BCD and 4CAD.
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Concept 1. Segments of Chords, Secants, and Tangents
m6 CDB = m6 BAC − m6 DBC
The measure of an Angle outside a circle is equal to half
the difference of the measures of the intercepted arcs
m6
BAC =
m6
ACD + m6
The measure of an exterior angle in a triangle equals
CDB
the sum of the measures of the remote interior angles
m6 CDB = m6 ACD + m6 CDB − m6 DBC
Combining the two steps above
m6 DBC = m6 ACD
algebra
Therefore, 4BCD ∼ 4CAD by the AA similarity postulate.
In similar triangles the ratios of corresponding sides are equal.
a
c
= ⇒ a(a + b) = c2 a+b c
Example 3
Find the value of the variable x assuming that it represents the length of a tangent segment.
x2 = 3(9 + 3)
x2 = 36
x=6
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Lesson Summary
In this section, we learned how to find the lengths of different segments associated with circles: chords, secants, and
tangents. We looked at cases in which the segments intersect inside the circle, outside the circle, or where one is
tangent to the circle. There are different equations to find the segment lengths, relating to different situations.
Review Questions
1. Find the value of missing variables in the following figures:
a.
b.
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Concept 1. Segments of Chords, Secants, and Tangents
c.
d.
e.
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f.
g.
h.
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Concept 1. Segments of Chords, Secants, and Tangents
i.
j.
9
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k.
l.
10
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Concept 1. Segments of Chords, Secants, and Tangents
m.
n.
o.
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p.
q.
r.
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Concept 1. Segments of Chords, Secants, and Tangents
s.
t.
2. A circle goes through the points A, B,C, and D consecutively. The chords AC and BD intersect at P. Given
that AP = 12, BP = 16, and CP = 6, find DP?
3. Suzie found a piece of a broken plate. She places a ruler across two points on the rim, and the length of the
chord is found to be 6 inches. The distance from the midpoint of this chord to the nearest point on the rim is
found to be 1 inch. Find the diameter of the plate.
4. Chords AB and CD intersect at P. Given AP = 12, BP = 8, and CP = 7, find DP.
Review Answers
1.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
14.4
16
4.5
32.2
12
29.67
4.4
18.03
4.54
20.25
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k. 7.48
l. 23.8
m. 24.4
n. 9.24 or 4.33
o. 17.14
p. 26.15
q. 7.04
r. 9.8
s. 4.4
t. 8
2. 4.5
3. 10 inches.
4. 13.71
Texas Instruments Resources
In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9694.
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