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Holt McDougal Geometry
12-6Segment Relationships in Circles
secant segmentexternal secant segmenttangent segment
Vocabulary
Holt McDougal Geometry
12-6Segment Relationships in Circles
In 1901, divers near the Greek island of Antikythera discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks.
The following theorem describes the relationship among the four segments that are formed when two chords intersect in the interior of a circle.
Holt McDougal Geometry
12-6Segment Relationships in Circles
Holt McDougal Geometry
12-6Segment Relationships in Circles
Example 1: Applying the Chord-Chord Product Theorem
Find the value of x and the length of each chord.
EJ JF = GJ JH
10(7) = 14(x)
70 = 14x
5 = x
EF = 10 + 7 = 17
GH = 14 + 5 = 19
J
Holt McDougal Geometry
12-6Segment Relationships in Circles
Check It Out! Example 1
Find the value of x and the length of each chord.
DE EC = AE EB
8(x) = 6(5)
8x = 30
x = 3.75
AB = 6 + 5 = 11
CD = 3.75 + 8 = 11.75
Holt McDougal Geometry
12-6Segment Relationships in Circles
Example 2: Art Application
The art department is contracted to construct a wooden moon for a play. One of the artists creates a sketch of what it needs to look like by drawing a chord and its perpendicular bisector. Find the diameter of the circle used to draw the outer edge of the moon.
Holt McDougal Geometry
12-6Segment Relationships in Circles
Example 2 Continued
8d = 145
8d – 64 = 81
8 (d – 8) = 9 9
Holt McDougal Geometry
12-6Segment Relationships in Circles
Check It Out! Example 2
What if…? Suppose the length of chord AB that the archeologists drew was 12 in. In this case how much longer is the disk’s diameter compared to the disk on p. 793?
AQ QB = PQ QR
6(6) = 3(QR)
12 = QR
12 + 3 = 15 = PR
6 in.
Holt McDougal Geometry
12-6Segment Relationships in Circles
A secant segment is a segment of a secant with at least one endpoint on the circle. An external secant segment is a secant segment that lies in the exterior of the circle with one endpoint on the circle.
Holt McDougal Geometry
12-6Segment Relationships in Circles
Holt McDougal Geometry
12-6Segment Relationships in Circles
Find the value of x and the length of each secant segment.
Example 3: Applying the Secant-Secant Product Theorem
16(7) = (8 + x)8
112 = 64 + 8x
48 = 8x
6 = x
ED = 7 + 9 = 16
EG = 8 + 6 = 14
Holt McDougal Geometry
12-6Segment Relationships in Circles
Check It Out! Example 3
Find the value of z and the length of each secant segment.
39(9) = (13 + z)13
351 = 169 + 13z
182 = 13z
14 = z
LG = 30 + 9 = 39
JG = 14 + 13 = 27
Holt McDougal Geometry
12-6Segment Relationships in Circles
A tangent segment is a segment of a tangent with one endpoint on the circle. AB and AC are tangent segments.
Holt McDougal Geometry
12-6Segment Relationships in Circles
Holt McDougal Geometry
12-6Segment Relationships in Circles
Example 4: Applying the Secant-Tangent Product Theorem
Find the value of x.
The value of x must be 10 since it represents a length.
ML JL = KL2
20(5) = x2
100 = x2
±10 = x
Holt McDougal Geometry
12-6Segment Relationships in Circles
Check It Out! Example 4
Find the value of y.
DE DF = DG2
7(7 + y) = 102
49 + 7y = 100
7y = 51
Holt McDougal Geometry
12-6Segment Relationships in Circles
Lesson Quiz: Part I
1. Find the value of d and the length of each chord.
d = 9
ZV = 17
WY = 18
2. Find the diameter of the plate.
Holt McDougal Geometry
12-6Segment Relationships in Circles
Lesson Quiz: Part II
x = 10
QP = 8
QR = 12
4. Find the value of a.
8
3. Find the value of x and the length of each secant segment.