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### Geometry BChapter 10

Segment Relationships in Circles

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

Use the lengths of segments in circles to solve problems.

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.

J discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks.

Example 1: Applying the Chord-Chord Product Theorem

Find the value of x and the length of each chord.

10(7) = 14(x)

70 = 14x

5 = x

EF = 10 + 7 = 17

GH = 14 + 5 = 19

In Your Notes! discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks. Example 1

Find the value of x and the length of each chord.

8(x) = 6(5)

8x = 30

x = 3.75

AB = 6 + 5 = 11

CD = 3.75 + 8 = 11.75

Example 2: Art Application discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks.

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 discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks.

8(d – 8) = 9 9

8d – 64 = 81

8d = 145

6 in. discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks.

In Your Notes! 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?

6(6) = 3(QR)

12 = QR

12 + 3 = 15 = PR

A discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks.secant segmentis a segment of a secant with at least one endpoint on the circle. An external secant segmentis a secant segment that lies in the exterior of the circle with one endpoint on the circle.

Example 3: Applying the Secant-Secant Product Theorem discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks.

Find the value of x and the length of each secant segment.

112 = 64 + 8x

48 = 8x

6 = x

ED = 7 + 9 = 16

EG = 8 + 6 = 14

In Your Notes! discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks. Example 3

Find the value of z and the length of each secant segment.

351 = 169 + 13z

182 = 13z

14 = z

LG = 30 + 9 = 39

JG = 14 + 13 = 27

A discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks.tangent segmentis a segment of a tangent with one endpoint on the circle. ABand ACare tangent segments.

Example 4: Applying the Secant-Tangent Product Theorem discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks.

Find the value of x.

ML JL = KL2

20(5) = x2

100 = x2

±10 = x

The value of x must be 10 since it represents a length.

In Your Notes! discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks. Example 4

Find the value of y.

DE DF = DG2

7(7 + y) = 102

49 + 7y = 100

7y = 51

Lesson Quiz: Part I discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks.

1. Find the value of d and the length of each chord.

2. Find the diameter of the plate.

Lesson Quiz: Part II discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks.

3. Find the value of x and the length of each secant segment.

4. Find the value of a.

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