Holt Geometry

11-6Segment Relationships in Circles

4

48

14

Warm UpSolve for x.

1.

2. 3x = 122

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

Holt Geometry

11-6Segment Relationships in Circles

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

Use the lengths of segments in circles to solve problems.

Objectives

Holt Geometry

11-6Segment 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 Geometry

11-6Segment Relationships in Circles

Holt Geometry

11-6Segment 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 Geometry

11-6Segment 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 Geometry

11-6Segment Relationships in Circles

Example 2: Art ApplicationThe 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.

8 (d – 8) = 9 9 Let d – 8 = x

8x = 81

x=81/8

8

81

8

64d

inind 125.188

118

8

145

Holt Geometry

11-6Segment Relationships in Circles

Example 2 Continued

8d = 145

8d – 64 = 81

OR 8 (d – 8) = 9 9

Holt Geometry

11-6Segment Relationships in Circles

Archaeologists discovered a fragment of an ancient disk.

To calculate its original diameter, they drew a chord AB and its perpendicular bisector PR.

The chord that the archeologists drew was 12 in. Find the disk’s diameter.

AQ QB = PQ QR

6(6) = 3(QR)

12 = QR

12 + 3 = 15 = PR

6 in.

Holt Geometry

11-6Segment 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 exactly one endpoint on the circle.

Holt Geometry

11-6Segment Relationships in Circles

Holt Geometry

11-6Segment 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 Geometry

11-6Segment 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 Geometry

11-6Segment 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 Geometry

11-6Segment Relationships in Circles

Holt Geometry

11-6Segment 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 Geometry

11-6Segment 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 Geometry

11-6Segment Relationships in Circles

Holt Geometry

11-6Segment Relationships in Circles

Homework

Page 795

Exercises 1-11

Holt Geometry

11-6Segment 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 Geometry

11-6Segment 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.