UNIT 12.4 SEGMENT LENGTHSUNIT 12.4 SEGMENT LENGTHS

4

48

14

Warm UpSolve for x.

1.

2. 3x = 122

3. BC and DC are tangent to A. Find BC.

Find the lengths of segments formed by lines that intersect circles.

Use the lengths of segments in circles to solve problems.

Objectives

secant segmentexternal secant segmenttangent segment

Vocabulary

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.

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 = 17GH = 14 + 5 = 19

J

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 = 11CD = 3.75 + 8 = 11.75

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.

Example 2 Continued

8d = 145

8d – 64 = 81

8 • (d – 8) = 9 • 9

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 = QR12 + 3 = 15 = PR

6 in.

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.

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

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

A tangent segment is a segment of a tangent with one endpoint on the circle. AB and AC are tangent segments.

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

Check It Out! Example 4

Find the value of y.

DE • DF = DG2

7(7 + y) = 102

49 + 7y = 100

7y = 51

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.

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.

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