Slope – Parallel and Perpendicular Lines

Geometry D – Chapter 3.3

Slope - Defined

Given two points (x1, y1) and (x2, y2), the slope m of a line is defined to be:

2 1

2 1

y y yorx x x

m

Slope – Example #1

Given two points A(-3, -4) and B(3, 5), find the slope of the line through the points.

2 1

2 1

y y yorx x x

m

Algebraically:

5 ( 4)3 ( 3)

m

5 4 93 3 6

Reduce fractions whenever possible.

32

m

Slope – Example #1

Given two points A(-3, -4) and B(3, 5), find the slope of the line through the points.

2 1

2 1

y y yorx x x

m

Graphically:

Graph the line through the points.Construct a right triangle.

Going from A to B, count the change in x and y.

A

B

6

9

9 36 2

ymx

Slope – Example #2a2 1

2 1

y y yorx x x

m

Algebraically, find the slope of the line AB.

A(-4, 4) B(1, 7)

7 4 31 ( 4) 5

m

The slope of the line through AB is 3/5.

Slope – Example #2b2 1

2 1

y y yorx x x

m

Graphically, find the slope of the line CD.

Going from C to D, the line goes down 3 and left 2.

-3

-2

3 32 2

m

The slope of the line is 3/2.

Slope – Example #2b2 1

2 1

y y yorx x x

m

The slope of line AB is 3/5.The slope of line CD is 3/2.

Compare line AB and line CD.What can you say about them?

The slope of line CD is greater than line AB. Line CD is steeper than line AB.

Slope – Example #2c2 1

2 1

y y yorx x x

m

On your own:

Find the slope of EF.Compare it with AB and CD.

The slope of EF is -2/3.

Since the slope is negative, the line runs down as we look from left to right. Positive slope runs up from left to right.Using the absolute values of the slopes. Therefore AB is not as steep as EF which is not as steep as CD.

3 2 35 3 2

Slope – Example #2d2 1

2 1

y y yorx x x

m

On your own:

Find the slope of CF.

C(3,7) F(3, -6)

6 7 133 3 0

m

Since division by zero is not defined, the slope of CF is undefined.

Slope – Vertical Lines2 1

2 1

y y yorx x x

m

The slope of any vertical line is undefined.

Slope – Example #32 1

2 1

y y yorx x x

m

On your own:

Find the slope of MN.

M(-1, 2) N(3, 2)

2 2 03 ( 1) 4

m

The slope of MN is 0. MN is said to have zero slope.

Slope – Horizontal Lines2 1

2 1

y y yorx x x

m

The slope of any horizontal line is zero.

Slope – Other Lines 2 1

2 1

y y yorx x x

m

Find the slope of PQ, RS, and TU.

Allow time for student work!

13

133

m of PQ

m of RS

m of TU

What can be said about lines PQ and RS?

The lines are parallel and have the same slope.

Slope – Parallel Lines2 1

2 1

y y yorx x x

m

Parallel lines have equal slopes.

Slope – Other Lines 2 1

2 1

y y yorx x x

m

13

133

m of PQ

m of RS

m of TU

What can be said about lines PQ and TU?

Use pages 1-2 of GSP file 3_3_pptdemo here!

The lines are perpendicular and have slopes which multiply to -1.

Slope – Perpendicular Lines2 1

2 1

y y yorx x x

m

Perpendicular lines have slopes which:

• multiply to -1

• are negative reciprocals of each other.

Horizontal and vertical lines are perpendicular to each other.

Slope – Example #42 1

2 1

y y yorx x x

m

Points R(3, -2) and S(-1, 3) form a line.

a) find the slope of RS.

3 ( 2) 5 541 3 4

m

b) find the slope of a line perpendicular to RS.

Since perpendicular lines have slopes which are negative reciprocals, the slope of the perpendicular line is 4/5.

Slope – Example #42 1

2 1

y y yorx x x

m

Points R(3, -2) and S(-1, 3) form a line.

c) knowing the slope of RS is -5/4, find another point on line RS.

The slope of -5/4 indicates a change of -5 in the y direction and 4 in the x direction.

X = 3 + 4 = 7Y = -2 + (-5) = -7Another point on the line is (7, --7)

Slope – Example #42 1

2 1

y y yorx x x

m

Points R(3, -2) and S(-1, 3) form a line.

Since we can also

move 5 units in the y direction and -4 units in the x direction. From Point S,

X = -1 + (-4) = -5Y = 3 + 5 = 8

Another point on the line is (-5, 8)

Can you find a similar point using S?5 5

4 4yx

Slope – Example #42 1

2 1

y y yorx x x

m

Points R(3, -2) and S(-1, 3) form a line.

d) knowing the slope of RS is -5/4, find the slope of the line perpendicular to RS.

Slopes of perpendicular lines are negative reciprocals of each other. The reciprocal of-5/4 is 4/5 or

5 51

4 4

Slope – Example #52 1

2 1

y y yorx x x

m

Points P(2, 5) and Q(4, y) form a line with a slope of 2/3. Find the value of y.

5 24 2 3y

5 22 3y

3 15 4y

3 19y

193

y