Lesson 9 - High Tech High
Lesson 9.8: Segments of Chords, Secants and Tangents
1
Objectives:
Find the lengths of segments associated with circles
Standards:
MATH.CA.8-12.GEOM.1.0, MATH.CA.8-12.GEOM.2.0, MATH.CA.8-12.GEOM.7.0, MATH.CA.8-12.GEOM.21.0
MATH.NCTM.9-12.GEOM.1.1, MATH.NCTM.9-12.GEOM.1.3
In this section we will discuss segments associated with circles and the angles formed by these segments. The figures below gives the names of segments associated with circles.
09.8.1 Theorem 9-14
9-14 Segments of Chords
If two chords intersect inside the circle then the segments of the chords satisfy the following relationship: [pic]
In words, this means that the product of the segments of one chord equals the product of segments of the second chord.
Proof:
We connect points A and C and points D and B to make triangles
[pic]and [pic].
[pic] vertical angles
[pic] inscribed angles intercepting the same arc
[pic] inscribed angles intercepting the same arc
Therefore [pic] by the AA similarity postulate.
In similar triangles the ratios of corresponding sides are equal.
[pic]
Example 1: Find the value of the variable.
Solution:
[pic]
09.8.2 Theorem 9-15
9-15 Segments of Secants
If two secants are drawn from a common point outside the circle then the segments of the secants satisfy the following relationship:
[pic]
In words, this means that the product of the outside segment of one secant and its whole length equals the product of the outside segment of the other secant and its whole length.
Proof:
We connect points A and D and points B and C to make triangles
[pic]and [pic].
[pic] same angle
[pic] inscribed angles intercepting the same arc
Therefore [pic] by the AA similarity postulate.
In similar triangles the ratios of corresponding sides are equal.
[pic]
This completes the proof.
Example 2: Find the value of the variable.
Solution:
[pic]
09.8.3 Theorem 9-16
9-16 Segments of Secants and Tangents
If a tangent and a secant are drawn from a point outside the circle then the segments of the secant and the tangent satisfy the following relationship: [pic]
In words, this means that the product of the outside segment of the secant and its whole length equals the square of the tangent segment.
Proof:
We connect points C and A and points B and C to make triangles
[pic]and [pic].
[pic] same angle
[pic] inscribed angles intercepting the same arc
Therefore [pic] by the AA similarity postulate.
In similar triangles the ratios of corresponding sides are equal.
[pic]
This completes the proof.
Example 3: Find the value of the variable.
Solution:
[pic]
Homework:
Find the value of missing variables in the following figures:
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
13. 14. 15.
16. 17.
18. 19.
20. 21. A circle goes through the points A, B, C, and D consecutively. The chords AC and BD intersect at P. Given that AP = 12, BP = 16, and CP = 6, how long is DP?
22. Suzie found a piece of a broken plate. She places a ruler across two points on the rim, and the length of the chord is found to be 6 inches. The distance from the midpoint of this chord to the nearest point on the rim is found to be 1 inch. Find the diameter of the plate.
23. Chords AB and CD intersect at P. Given AP = 12, BP = 8, and CP = 7, find DP.
Answers:
1. 14.4 2. 16 3. 4.5 4. 32.2
5. 12 6. 29.67 7. 4.4 8. 18.03
9. 4.54 10. 20.25 11. 7.48 12. 23.8
13. 24.4 14. 9.24 or 4.33 15. 17.14 16. 26.15
17. 7.04 18. 9.8 19. 4.4 20. 6.24
21. 4.5 22. 10 inches. 23. 13.71 [pic]
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8
9
5
x
20
x
8
10
50
x
15
15
25
x
18
18
x
8
10
24
x
20
30
10
12
45
x
x
2
20
3x
5
x – 3
x + 3
secant segment
external segment
of a secant
tangent segment
segment of a chord
a
[pic]b
c
d
a
b
c
d
a
b
c
10
x
[pic]12
8
x
10
20
9
x
9
3
x
6
5
25
14
3x
12
8
x
5
5
x
10
9
x
7
8
4
x
8
x
5
10
7
x
10
12
x
12
5
8
40
x
15
35
6
15
z + 3
2z + 1
5z + 2
3z – 5
d
c
[pic]b
a
A
B
C
D
E
d
c
b
a
A
B
C
D
N
D
C
B
A
c
b
a
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