Post on 27-Mar-2015
Geometry
10.5 Segment Length in Circles
mbhaub@mpsaz.org
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 2
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
A quadratic equation is in the form
Any quadratic equation can be solved using the formula
2 0ax bx c
2 4
2
b b acx
a
Skip
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 4
Solve: 2 2 15 0x x
2
2
4
2
2 2 4(1)( 15) 2 4 60
2(1) 2
2 64 2 8
2 2
b b acx
a
x
x
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 5
Solve: 2 2 15 0x x
2
2
4
2
2 2 4(1)( 15) 2 4 60
2(1) 2
2 64 2 8
2 2
b b acx
a
x
x
2 8 6
32 22 8 10
52 2
x
x
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 6
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
a
c
d
b
a b = c d
Theorem Demo
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 8
Example 1 Find a.
10
48
10 4 = 8 a
40 = 8 a
5 = a5a
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Geometry 10.5 Segment Lengths in Circles 9
Your Turn: Find x.
A
B
C
D
E3x
x
8
3x x = 8 6
3x2 = 48
x2 = 16
x = 4
6
4
12
Check: 12 4 = 48 and 8 6 = 48
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 10
Proof
R
S
O
T
V
3
41 2
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.
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 11
Proof continued
R
S
O
T
V
3
41 2
Now SOR ~ VOT. Why?
AA~ Postulate.
In similar triangles, sides are proportional:
OR OT=
OS OV
OR OV = OT OS
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 12
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
Terminology
This segment is the external secant segment.
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 14
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
Terminology
A
B
C
D
AC is a __________________.
AB is the _________________________.
AD is a _________________.
external secant segment
secant segment
tangent segment
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 16
Theorem 10.17 (tangent-secant)
A
B
C
D
2AD AB AC
Theorem Demo
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 17
Theorem 10.17 (simplified)
a
b
c
c2 = a(a + b)
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 18
Example 2 Find AD.
A
B
C
D
6
4
2 4(4 6)
4(10)
40
4 10
2 10
AD
AD
AD
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 19
Your Turn. Solve for x.
x
4
4
2 4(8)
32
32
16 2
4 2
x
x
x
8
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 20
Turn it up a notch…
x
5
4
2
2
5 ( 4)
25 4
x x
x x
Now What?
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 21
Quadratic Equation
2
2
5 ( 4)
25 4
x x
x x
Set quadratic equations equal to zero.2 4 25 0x x
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 22
Quadratic Formula
2 42
b b acx
a
2 4 25 0x x 1a = 1 b = 4 c = -25
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 23
Quadratic Formula
2 1 ( 254 4
1)
24
x
2 4 25 0x x 1a = 1 b = 4 c = -25
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 24
Solve it.
24 4 4(1)( 25)2(1)
4 16 1002
4 1162
x
4 10.772
6.773.39
24 10.77
214.77
27.39
x
x
x can’t be negative
x 3.39
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 25
Just do it!
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 26
Your Turn Solve for x.
Equation:
32 = x(x + 2)
3
2
x
x + 2
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 27
Solution
3
2 x
32 = x(x + 2)
9 = x2 + 2x
0 = x2 + 2x – 9
a = 1
b = 2
c = -9
22 2 4(1)( 9)2(1)
2 4 362
2 402
2 6.322
42.16
.322
x
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 28
Theorem 10.16 (secant-secant)
a
b
c d
a(a+b) = c(c+d)
Theorem Demo
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 29
Example 3 Solve for x.
5
8
6X
Solution:
5(5 + 8) = 6(6 + x)
5(13) = 36 + 6x
65 = 36 + 6x
29 = 6x
x = 4 5/6 (or 4.83)
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 30
Your Turn Solve for x.
9
11
10X
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 31
Solution
9
11
10X
9(9 11) 10(10 )
9(20) 10 100
180 10 100
8 1
8
0 0
x
x
x
x
x
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 32
Example 4 Solve for x.
16
12
5
X
Equation:
5x = 4(16)
Why?
5x = 64
x = 12.8
4
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 33
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
Formula Summary
c
a
b
c2 = a(a + b)
ac
d
b
ab = cd
a(a+b) = c(c+d)
ab
c d
Tuesday, March 24, 2:56
Geometry 10.5 Segment Lengths in Circles 35
Homework
mbhaub@mpsaz.org