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Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

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Page 1: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

Holt McDougal Geometry

12-6Segment Relationships in Circles

secant segmentexternal secant segmenttangent segment

Vocabulary

Page 2: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent 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.

Page 3: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

Holt McDougal Geometry

12-6Segment Relationships in Circles

Page 4: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

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

Page 5: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

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

Page 6: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

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.

Page 7: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

Holt McDougal Geometry

12-6Segment Relationships in Circles

Example 2 Continued

8d = 145

8d – 64 = 81

8 (d – 8) = 9 9

Page 8: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

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.

Page 9: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

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.

Page 10: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

Holt McDougal Geometry

12-6Segment Relationships in Circles

Page 11: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

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

Page 12: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

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

Page 13: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

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.

Page 14: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

Holt McDougal Geometry

12-6Segment Relationships in Circles

Page 15: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

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

Page 16: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

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

Page 17: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

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.

Page 18: Holt McDougal Geometry 12-6 Segment Relationships in Circles secant segment external secant segment tangent segment Vocabulary

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.