geometry section 3-3 1112

Section 3-3Slopes of Lines

Monday, December 19, 2011

Essential Questions

How do you find slopes of lines?

How do you use slope to identify parallel and perpendicular lines?


Vocabulary

1. Slope:

2. Rate of Change:


Vocabulary

1. Slope: The ratio of the vertical change to the horizontal change between two points

2. Rate of Change:


Vocabulary

1. Slope: The ratio of the vertical change to the horizontal change between two points; Change in y over change in x

2. Rate of Change:


Vocabulary


rise over run

2. Rate of Change:


Vocabulary


rise over run

2. Rate of Change: A way to describe slope


ExploreGraph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between

the horizontal lines.

Distance between vertical

Distance between horizontal



the horizontal lines.y

x






x

A






x

A

B






x

A

B

Distance between vertical5 Units





x

A

B


Distance between horizontal7 Units




x

A

B


Distance between horizontal7 Units

Up 7, Right 5


Slope Formula


Slope Formula

m =y2 − y1x2 − x1

for points

(x1,y1),(x2,y2 )


Example 1Find the slope of the line that goes

through the following pairs of points.

a. C(−3, 4) and D(8, 1)




a. C(−3, 4) and D(8, 1)(x1,y1)




a. C(−3, 4) and D(8, 1)(x1,y1) (x2,y2 )




a. C(−3, 4) and D(8, 1)(x1,y1) (x2,y2 )

m =y2 − y1x2 − x1




a. C(−3, 4) and D(8, 1)(x1,y1) (x2,y2 )

m =y2 − y1x2 − x1

=1− 48 − (−3)




a. C(−3, 4) and D(8, 1)(x1,y1) (x2,y2 )

m =y2 − y1x2 − x1

=1− 48 − (−3)

=−311




a. C(−3, 4) and D(8, 1)(x1,y1) (x2,y2 )

m =y2 − y1x2 − x1

=1− 48 − (−3)

=−311

Down 3, Right 11




b. E(5, −1) and F(−3, 7)




b. E(5, −1) and F(−3, 7)

m =y2 − y1x2 − x1




b. E(5, −1) and F(−3, 7)

m =y2 − y1x2 − x1

=7 − (−1)−3 − 5




b. E(5, −1) and F(−3, 7)

m =y2 − y1x2 − x1

=7 − (−1)−3 − 5

=8−8




b. E(5, −1) and F(−3, 7)

m =y2 − y1x2 − x1

=7 − (−1)−3 − 5

=8−8

= −1




b. E(5, −1) and F(−3, 7)

m =y2 − y1x2 − x1

=7 − (−1)−3 − 5

=8−8

= −1

Down 1, Right 1




c. G(−1, 2) and H(−1, 7)




c. G(−1, 2) and H(−1, 7)

m =y2 − y1x2 − x1




c. G(−1, 2) and H(−1, 7)

m =y2 − y1x2 − x1

=7 − 2

−1− (−1)




c. G(−1, 2) and H(−1, 7)

m =y2 − y1x2 − x1

=7 − 2

−1− (−1)=50




c. G(−1, 2) and H(−1, 7)

m =y2 − y1x2 − x1

=7 − 2

−1− (−1)=50

Undefined




c. G(−1, 2) and H(−1, 7)

m =y2 − y1x2 − x1

=7 − 2

−1− (−1)=50

Up 5, Right 0

Undefined




c. G(−1, 2) and H(−1, 7)

m =y2 − y1x2 − x1

=7 − 2

−1− (−1)=50

Up 5, Right 0

Undefined

Vertical LineMonday, December 19, 2011



d. J(3, 4) and K(−2, 4)




d. J(3, 4) and K(−2, 4)

m =y2 − y1x2 − x1




d. J(3, 4) and K(−2, 4)

m =y2 − y1x2 − x1

=4 − 4−2 − 3




d. J(3, 4) and K(−2, 4)

m =y2 − y1x2 − x1

=4 − 4−2 − 3

=0−5




d. J(3, 4) and K(−2, 4)

m =y2 − y1x2 − x1

=4 − 4−2 − 3

=0−5

Up 0, Left 5




d. J(3, 4) and K(−2, 4)

m =y2 − y1x2 − x1

=4 − 4−2 − 3

=0−5

Up 0, Left 5 Horizontal Line




d. J(3, 4) and K(−2, 4)

m =y2 − y1x2 − x1

=4 − 4−2 − 3

=0−5

Up 0, Left 5 Horizontal Line

= 0


Example 2In 2000, the annual sales for one

manufacturer of camping equipment was $48.9 million. In 2005, the total sales

were $85.9 million. If sales increase at the same rate, what will the total sales

be in 2015?





be in 2015?

85.9 − 48.9 =





be in 2015?

85.9 − 48.9 = 37





be in 2015?

85.9 − 48.9 = 37Every 5 years, sales increase by $37 million





be in 2015?


85.9 + 2(37) =





be in 2015?


85.9 + 2(37) = 159.9





be in 2015?


85.9 + 2(37) = 159.9

In 2015, sales should be about $159.9 million


PostulatesSlopes of parallel lines:

Slopes of perpendicular lines:



Two lines will be parallel IFF they have the same slope. All vertical lines

are parallel.




Two lines will be parallel IFF they have the same slope. All vertical lines

are parallel.

Two lines will be perpendicular IFF the product of their slopes is −1. Vertical

and horizontal lines are perpendicular.



Example 3Determine whether FG and HJ are

parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3).

Graph each line to verify your answer.





m(FG

) = y2 − y1x2 − x1





m(FG

) = y2 − y1x2 − x1

=−1− (−3)−2 −1





m(FG

) = y2 − y1x2 − x1

=−1− (−3)−2 −1

=2−3





m(FG

) = y2 − y1x2 − x1

=−1− (−3)−2 −1

=2−3

m(HJ ) = y2 − y1

x2 − x1





m(FG

) = y2 − y1x2 − x1

=−1− (−3)−2 −1

=2−3

m(HJ ) = y2 − y1

x2 − x1=3 − 06 − 5





m(FG

) = y2 − y1x2 − x1

=−1− (−3)−2 −1

=2−3

m(HJ ) = y2 − y1

x2 − x1=3 − 06 − 5

=31





m(FG

) = y2 − y1x2 − x1

=−1− (−3)−2 −1

=2−3

m(HJ ) = y2 − y1

x2 − x1=3 − 06 − 5

=31= 3





m(FG

) = y2 − y1x2 − x1

=−1− (−3)−2 −1

=2−3

m(HJ ) = y2 − y1

x2 − x1=3 − 06 − 5

=31= 3

Neither


y

x

Example 4Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4)

and N(2, 1).


y

x


and N(2, 1).

m =y2 − y1x2 − x1


y

x


and N(2, 1).

m =y2 − y1x2 − x1

=4 −1−2 − 2


y

x


and N(2, 1).

m =y2 − y1x2 − x1

=4 −1−2 − 2

=3−4


y

x


and N(2, 1).

m =y2 − y1x2 − x1

=4 −1−2 − 2

=3−4

m = −34


y

x


and N(2, 1).

m =y2 − y1x2 − x1

=4 −1−2 − 2

=3−4

m = −34

Q


y

x


and N(2, 1).

m =y2 − y1x2 − x1

=4 −1−2 − 2

=3−4

m = −34

QN

M


Check Your Understanding

Use problems 1-11 to check the ideas from this lesson


Problem Set


Problem Set

p. 191 #13-39 odd

“I have found power in the mysteries of thought.” - Euripides


geometry section 3-3 1112

Education