1. 10.5 Segment Length in Circles
Geometry
10.5 Segment Length in Circles

2. Geometry 10.5 Segment Lengths in Circles
Goals
Find the lengths of segments of chords.
Find the lengths of segments and tangents.
2-Day Lesson
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

3. Quadratic Equation Review
Skip
A quadratic equation is in the form
Any quadratic equation can be solved using the formula
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

4. Geometry 10.5 Segment Lengths in Circles
Solve:
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

5. Geometry 10.5 Segment Lengths in Circles
Solve:
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

6. Geometry 10.5 Segment Lengths in Circles
10.5 Chords in a Circle
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

7. Chords in a Circle Theorem 10.15
Theorem Demo d a b c
a  b = c  d
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

8. Geometry 10.5 Segment Lengths in Circles
Example 1 Find a.
10  4 = 8  a
40 = 8  a
5 = a 10 5 a 4 8
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

9. Geometry 10.5 Segment Lengths in Circles
Your Turn: Find x.
3x  x = 8  6
3x2 = 48
x2 = 16
x = 4 A D 3x 6 12 E x 4 B 8 C
Check: 12  4 = 48 and 8  6 = 48
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

10. Geometry 10.5 Segment Lengths in Circles
Proof
3 and 4 both intercept arc SV. What does this tell use about 3 and 4?
They are congruent.
What kind of angles are 1 and 2?
Vertical Angles
And vertical angles are ____.
Congruent. R T 3 O 4 1 2 V S
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

11. Geometry 10.5 Segment Lengths in Circles
Proof continued
Now SOR ~ VOT. Why?
AA~ Postulate.
In similar triangles, sides are proportional: R T 3 O 4 1 2 V
OR  OV = OT  OS S
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

12. Geometry 10.5 Segment Lengths in Circles
Terminology
This line is a secant.
This segment is a secant segment.
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

13. Geometry 10.5 Segment Lengths in Circles
Terminology
This segment is the external secant segment.
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

14. Geometry 10.5 Segment Lengths in Circles
Terminology
This line is a tangent.
This segment is a tangent segment.
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

15. Geometry 10.5 Segment Lengths in Circles
Terminology
AC is a __________________.
AB is the _________________________.
AD is a _________________.
secant segment
external secant segment
tangent segment A B C D
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

16. Theorem 10.17 (tangent-secant)B C D
Theorem Demo
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Geometry 10.5 Segment Lengths in Circles

17. Geometry 10.5 Segment Lengths in Circles
Theorem (simplified)
c2 = a(a + b) b a c
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

18. Geometry 10.5 Segment Lengths in Circles
Example 2 Find AD. A B C D 6 4
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

19. Geometry 10.5 Segment Lengths in Circles
Your Turn. Solve for x. 8 4 4 x
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

20. Geometry 10.5 Segment Lengths in Circles
Turn it up a notch… 4 x
Now What? 5
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

21. Geometry 10.5 Segment Lengths in Circles
Quadratic Equation
Set quadratic equations equal to zero.
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

22. Geometry 10.5 Segment Lengths in Circles
Quadratic Formula 1
a = 1
b = 4
c = -25
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

23. Geometry 10.5 Segment Lengths in Circles
Quadratic Formula 1
a = 1
b = 4
c = -25
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

24. Geometry 10.5 Segment Lengths in Circles
Solve it.
x can’t be negative
x  3.39
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

25. Geometry 10.5 Segment Lengths in Circles
All that for just
one problem?
Just do it!
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

26. Geometry 10.5 Segment Lengths in Circles
Your Turn Solve for x. 3 2 x
Equation:
32 = x(x + 2)
x + 2
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

27. Geometry 10.5 Segment Lengths in Circles3 2 x
Solution
32 = x(x + 2)
9 = x2 + 2x
0 = x2 + 2x – 9
a = 1
b = 2
c = -9
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

28. Theorem 10.16 (secant-secant)
Theorem Demo b a d c
a(a+b) = c(c+d)
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

29. Geometry 10.5 Segment Lengths in Circles
Example 3 Solve for x.
Solution:
5(5 + 8) = 6(6 + x)
5(13) = x
65 = x
29 = 6x
x = 4 5/6 (or 4.83) 5 8 6 X
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

30. Geometry 10.5 Segment Lengths in Circles
Your Turn Solve for x. 9 11 10 X
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

31. Geometry 10.5 Segment Lengths in Circles
Solution 9 11 10 X
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

32. Geometry 10.5 Segment Lengths in Circles
Example 4 Solve for x.
Equation:
5x = 4(16)
Why?
5x = 64
x = 12.8 16 12 5 X 4
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

33. Geometry 10.5 Segment Lengths in Circles
Summary
Segments in a circle have three situations:
Chord-Chord
Secant-Tangent
Secant-Secant
Do you know the formula for each?
Read the problems carefully. Use the correct numbers for each variable.
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles

34. Geometry 10.5 Segment Lengths in Circles
Formula Summary
a(a+b) = c(c+d) a b c d c a b
c2 = a(a + b) a c d b
ab = cd
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Geometry 10.5 Segment Lengths in Circles

35. Geometry 10.5 Segment Lengths in Circles
Homework
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles