Geometry

3.7 Perpendicular Lines in the Coordinate Plane

April 19, 2023 Geometry 3.7 Perpendicular Lines in the Coordinate Plane 2

Goals

Use slope to identify perpendicular lines in a coordinate plane.

Write equations of perpendicular lines.

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Review Lines are parallel if they have the same

slope. Slope is rise/run. Lines with a positive slope rise to the right. Lines with a negative slope rise to the left. Horizontal Lines: slope = 0. Vertical Lines: slope is undefined (or none).

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Problem SlopesFind the slope of the line containing (4, 6) and (2, 6).

02

0

24

66

m

Do it graphically:

(2, 6) (4, 6)

y = 6Horizontal Lines have the form y = c.

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Problem SlopeFind the slope of the line containing (4, 6) and (4, 3).

0

3

44

36m

Do it graphically:

(4, 6)

(4, 3)

x = 4vertical Lines have the form x = c.

undefined (no SlopE)

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Postulate 18Two lines are perpendicular iff the product of their slopes is –1.

Algebraically: m1 • m2 = –1

A vertical and a horizontal line are perpendicular.

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Example

1

2m1 2

11 m

m2

2

-122 m

122

121 mm

m1 m2

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You don’t need a picture.Line A contains (2, 7) and (4, 13).

Line B contains (3, 0) and (6, -1).

Are the lines perpendicular?

32

6

24

713

Am3

1

36

01

Bm

13

1)3(

YES!

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April 19, 2023 Geometry 3.7 Perpendicular Lines in the Coordinate Plane 10

Exception

(2, 2)

(2, -1)

(3, 1)(-2, 1)

m1

m2

Slope of m1 is ?

Undefined

Slope of m2 is ?

Zero

m1 m2 –1.

But m1 m2!A vertical line and a horizontal line are perpendicular.

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Another way to think of it:Two lines are perpendicular if one slope is the negative reciprocal of the other.

3

5

5

3

8

18

99

1

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Slope Intercept form review

y = mx + b m is the slope b is the y-intercept The y-intercept is at (0, b) Lines are parallel if they have the same

slope. They are perpendicular if the product of their slopes is –1.

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More challenging problem

232 :

223 :

2

1

yxp

yxp

These equations are in General Form

Ax + By = C

Slope is always:B

A

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Why is this so?

Consider the equation: 8x – 4y = 12

Move the 8x: – 4y = – 8x + 12

Divide by –4: y = 2x – 3

Slope is? 2

Now use –A/B: -8/(-4) = 2

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For –3x + 2y = 2, slope is2

3

2

3

B

A

For 2x + 3y = –2, slope is3

2

B

A

The slopes are negative reciprocals, so the lines are perpendicular.

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In summary

Two lines are parallel if they have the same slope.

Two lines are perpendicular if the product of their slopes is –1.

General form is Ax + By = C and the slope in this form is –A/B.

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Homework