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Example 1 : The two lines given below are perpendicular. What do you
notice about the slopes of those two lines?
๐
and ๐
๐
have slopes that are the negative reciprocal of each
other (๐
๐
๐
๐
๐
๐
๐
). This is true not just for these two
lines which are perpendicular, but for all perpendicular lines.
Perpendicular lines:
- two lines that intersect at a right angle (90ยฐ)
- the slopes of the lines are the negative reciprocal of each other
(opposite signs, flipped over)
o if two lines are perpendicular, and the slope of the first line is ๐,
the slope of the second line is โ
1
๐
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4 5
Two points
that ๐
1
passes
through:
( 0 , โ 2 ), ( 3 , 2 )
Slope of ๐
1
:
๐ =
โ๐ฆ
โ๐ฅ
=
4
3
Two points
that ๐
2
passes
through:
( โ 4 , 4
) ,
( 0 , 1
)
Slope of ๐
2
:
๐ =
โ๐ฆ
โ๐ฅ
=
โ 3
4
To find the
slope of ๐
1
(the blue
line) or ๐
2
(the red line),
simply pick
any two
points that ๐
1
passes
through, and
then find the
vertical
change
divided by
the
horizontal
change
(๐ =
โ๐ฆ
โ๐ฅ
)
๐
1
๐
2
Example 2: Find the slope of a line that is perpendicular to the line
2
3
๐ฅ โ 4. Enter exact answers only (no approximations).
When a linear equation is expressed in slope-intercept form, like
2
3
๐ฅ โ 4 , the slope is simply the coefficient of ๐ฅ. So in this case the
slope is
2
3
In order to be perpendicular to the line ๐ฆ =
2
3
๐ฅ โ 4 , any other line must
have a slope that is the negative reciprocal of that. So perpendicular lines
must have a slope of โ
Example 3 : Find the slope of a line that is perpendicular to the line
9 ๐ฅ โ 4 ๐ฆ = 5. Enter exact answers only (no approximations).
Example 6 : Find the equation of the line that passes through the point
๐ด( 7 , โ 3 ) and is perpendicular to the line 2 ๐ฅ โ 5 ๐ฆ = 8. Enter exact
answers only (no approximations), and write the equation in slope-
intercept form (๐ฆ = ๐๐ฅ + ๐), if possible.
Example 7 : Find the equation of the line with a ๐ฆ-intercept of 2 , that is
perpendicular to the ๐ฆ-axis. Enter exact answers only (no
approximations), and write the equation in slope-intercept form
(๐ฆ = ๐๐ฅ + ๐), if possible.
Keep in mind that horizontal lines have a slope of zero and are of the form
๐ฆ = #, while vertical lines have an undefined slope and are of the form
The ๐ฆ-axis is a vertical line. A line
perpendicular to the ๐ฆ-axis would be a
horizontal line. That means its slope would
be zero.
Example 8 : Find the equation of the line with an ๐ฅ-intercept of 5 , that is
perpendicular to the line 5 ๐ฅ + 9 ๐ฆ =
1
8
. Enter exact answers only (no
approximations), and write the equation in slope-intercept form
, if possible.
On this problem Iโll start by converting the linear equation 5 ๐ฅ + 9 ๐ฆ =
1
8
from general form to slope-intercept form so I can identify its slope.
Now that I know the slope of the given line is โ
5
9
, I know that in order for
another line to be perpendicular to the given line, its slope must be the
negative reciprocal of that (
9
5
). So I now have the slope of the line Iโm
trying to find (๐ =
9
5
). And since the line Iโm trying to find has an ๐ฅ-
intercept of 5 , I also have a point that I can use ( 5 , 0 ). So now Iโll plug
this information into point-slope form:
1
1
