Special Segments in a Circle

Chapter 10.7

• Find measures of segments that intersect in the interior of a circle.

• Find measures of segments that intersect in the exterior of a circle.

Standard 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. (Key)

Standard 21.0 Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles. (Key)

Chord Segment Theorem• If two chords intersect in the interior of a circle, then the product of

the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

• Forget the words, copy the picture.

5 cm

4 cm

2 cm

10 cm

B

E

D

A

C

(AB)(BC) = (DB)(BE)

(2)(10) = (4)(5)

20 = 20

Intersection of Two Chords

Find x.

Answer: 13.5

A. A

B. B

C. C

D. D

0% 0%0%0%

A. 14

B. 12.5

C. 2

D. 18

Find x.

Example: Solve for x6(x + 2) = 3(3x – 1)

6x + 12 = 9x – 3

15 = 3x

5 = x

x + 2 cm

6 cm

3 cm

3x - 1 cm

B

E

D

A

C

BIOLOGY Biologists often examine organisms under microscopes. The circle represents the field of view under the microscope with a diameter of 2 mm. Determine the length of the organism if it is located 0.25 mm from the bottom of the field of view. Round to the nearest hundredth.

1.75

x x

4375.x

1. A

2. B

3. C

4. D

0%0%0%0%

A B C D

A. 10 ft

B. 20 ft

C. 36 ft

D. 18 ft

ARCHITECTURE Phil is installing a new window in an addition for a client’s home. The window is a rectangle with an arched top called an eyebrow. The diagram below shows the dimensions of the window. What is the radius of the circle containing the arc if the eyebrow portion of the window is not a semicircle?

Hint:

Secant Segment TheoremSecant Segment Theorem• If two secant segments share the same endpoint outside a circle,

then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment.

• Forget the words, copy the picture.

D

BA C

E

(AB)(AC) = (AD)(AE)

Example: Solve for x(9)(20) = (10)(10 + x)

180 = 100 + 10x

80 = 10x

8 = x11

10

x

9

D

BA C

E

20

10 + x

Intersection of Two SecantsFind x if EF = 10, EH = 8, and FG = 24.

Answer: 34.5

1. A

2. B

3. C

4. D

0%0%0%0%

A B C D

A. 28.125

B. 50

C. 26

D. 28

Find x if GO = 27, OM = 25, and IK = 24.

Secant-Tangent Segment TheoremSecant-Tangent Segment Theorem• If a secant segment and a tangent segment share an endpoint

outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.

• Forget the words, copy the picture.

(AC)(AD) = (AB)2

C

D

B

A

Example: Solve for x(6)(6 + x) = (12)2

36 + 6x = 144

6x = 108

x = 18

6 + x

x

6

12

C

D

B

A

Find x. Assume that segments that appear to be tangent are tangent.

Answer: 8

Intersection of a Secant and a Tangent

Disregard the negative solution.

A. A

B. B

C. C

D. D

0% 0%0%0%

A. 22.36

B. 25

C. 28

D. 30

Find x. Assume that segments that appear to be tangent are tangent.

Homework Chapter 10.7• Pg 611

5 – 15, 17 – 24, 41 – 43