Geometry 10.5 Segment Length in Circles [email protected].

Geometry

10.5 Segment Length in Circles

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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



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



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



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



10.5 Chords in a Circle



Chords in a Circle Theorem 10.15

a

c

d

b

a b = c d

Theorem Demo



Example 1 Find a.

10

48

10 4 = 8 a

40 = 8 a

5 = a5a



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



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.



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



Terminology

This line is a secant.

This segment is a secant segment.



Terminology

This segment is the external secant segment.



Terminology

This line is a tangent.

This segment is a tangent segment.



Terminology

A

B

C

D

AC is a __________________.

AB is the _________________________.

AD is a _________________.

external secant segment

secant segment

tangent segment



Theorem 10.17 (tangent-secant)

A

B

C

D

2AD AB AC

Theorem Demo



Theorem 10.17 (simplified)

a

b

c

c2 = a(a + b)



Example 2 Find AD.

A

B

C

D

6

4

2 4(4 6)

4(10)

40

4 10

2 10

AD

AD

AD



Your Turn. Solve for x.

x

4

4

2 4(8)

32

32

16 2

4 2

x

x

x

8



Turn it up a notch…

x

5

4

2

2

5 ( 4)

25 4

x x

x x

Now What?



Quadratic Equation

2

2

5 ( 4)

25 4

x x

x x

Set quadratic equations equal to zero.2 4 25 0x x



Quadratic Formula

2 42

b b acx

a

2 4 25 0x x 1a = 1 b = 4 c = -25



Quadratic Formula

2 1 ( 254 4

1)

24

x

2 4 25 0x x 1a = 1 b = 4 c = -25



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



Just do it!



Your Turn Solve for x.

Equation:

32 = x(x + 2)

3

2

x

x + 2



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



Theorem 10.16 (secant-secant)

a

b

c d

a(a+b) = c(c+d)

Theorem Demo



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)



Your Turn Solve for x.

9

11

10X



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



Example 4 Solve for x.

16

12

5

X

Equation:

5x = 4(16)

Why?

5x = 64

x = 12.8

4



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.



Formula Summary

c

a

b

c2 = a(a + b)

ac

d

b

ab = cd

a(a+b) = c(c+d)

ab

c d



Homework

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Geometry 10.5 Segment Length in Circles [email protected].

Documents

segment lengths

x slide

b slide

b c d slide

lengths of segments

circle slide

x equation

x solution