Perpendicular Lines

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Geometry
3.7 Perpendicular Lines in
the Coordinate Plane
Goals
Use slope to identify perpendicular lines
in a coordinate plane.
 Write equations of perpendicular lines.

April 10, 2015
Geometry 3.7 Perpendicular Lines in the Coordinate Plane
2
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). Geometry 3.7 Perpendicular Lines in the Coordinate Plane
3

April 10, 2015
Problem Slopes
Find the slope of the line containing (4, 6)
and (2, 6).
Do it graphically:
m 
66
42

0
0
2
Horizontal Lines have the
form y = c.
April 10, 2015
(2, 6)
(4, 6)
y=6
Geometry 3.7 Perpendicular Lines in the Coordinate Plane
4
Problem Slope
Find the slope of the line containing (4, 6)
and (4, 3).
Do it graphically:
m 
63
44

3

0
undefined (no SlopE)
vertical Lines have the form
x = c.
April 10, 2015
(4, 6)
(4, 3)
x=4
Geometry 3.7 Perpendicular Lines in the Coordinate Plane
5
Postulate 18
Two lines are perpendicular iff the product
of their slopes is –1.
Algebraically: m1 • m2 = –1
A vertical and a horizontal line are
perpendicular.
April 10, 2015
Geometry 3.7 Perpendicular Lines in the Coordinate Plane
6
Example
m1
1
m1 
2
2
m2  2
1
-1
m1  m 2 
2
1
2
  2    1
m1  m2
m2
April 10, 2015
Geometry 3.7 Perpendicular Lines in the Coordinate Plane
7
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?
mA 
13  7
42

6
3
2
 1
(3)     1
 3
April 10, 2015
mB 
1 0
63

1
3
YES!
Geometry 3.7 Perpendicular Lines in the Coordinate Plane
8
April 10, 2015
Geometry 3.7 Perpendicular Lines in the Coordinate Plane
9
Exception
Slope of m1 is ?
 Undefined
m1
Slope of m2 is ?
(2, 2)
m2
(-2, 1)
(3, 1)
(2, -1)
 Zero
m1  m2  –1.
But m1  m2!
A vertical line and a
horizontal line are
perpendicular.
April 10, 2015
Geometry 3.7 Perpendicular Lines in the Coordinate Plane
10
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
1
8
1
 9
9
April 10, 2015
Geometry 3.7 Perpendicular Lines in the Coordinate Plane
11
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.

April 10, 2015
Geometry 3.7 Perpendicular Lines in the Coordinate Plane
12
More challenging problem
p1 :  3 x  2 y  2
p2 :
2 x  3 y  2
These equations are in General Form
Ax + By = C
Slope is always: 
A
B
April 10, 2015
Geometry 3.7 Perpendicular Lines in the Coordinate Plane
13
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
April 10, 2015
Geometry 3.7 Perpendicular Lines in the Coordinate Plane
14
For –3x + 2y = 2, slope is 
For 2x + 3y = –2, slope is 
A

B
A
B
3
2


3
2
2
3
The slopes are negative reciprocals, so
the lines are perpendicular.
April 10, 2015
Geometry 3.7 Perpendicular Lines in the Coordinate Plane
15
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.

April 10, 2015
Geometry 3.7 Perpendicular Lines in the Coordinate Plane
16
Homework
April 10, 2015
Geometry 3.7 Perpendicular Lines in the Coordinate Plane
17
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