4.5A Find and Use
Slopes of Lines
Recall: The slope of a non-vertical line is the
ratio of vertical change (rise) to horizontal
change (run) between any two points on the
line.
Slope is rise over run:
1
2/8 =
4
Up 2
Right 8
𝑦2 − 𝑦1
𝑚=
𝑥2 − 𝑥1
𝑚=
4 −2
5 −−3
2
8
= =
𝟏
𝟒
Slope is rise over run:
−𝟐
-4/6 =
𝟑
𝑦2 − 𝑦1
𝑚=
𝑥2 − 𝑥1
𝑚=
down 4
Right 6
−4 −0
2 −−4
=
−4
6
=
−𝟐
𝟑
Slopes of Intersecting Lines - The steeper line has the slope with _greater absolute
value___.
Slopes of Parallel Lines – In a coordinate plane, two non-vertical lines are parallel if
and only if __they have the same slope_____________________________
Any two __vertical__ lines are parallel.
Slopes of Perpendicular Lines – In a coordinate plane, two non-vertical lines are
perpendicular if and only if the product of their slopes is _-1__ (or their slopes are
_negative reciprocals__).
Vertical and horizontal lines are perpendicular.
𝑚𝑗 =
0 −6
2 −0
=
−6
2
= −3
1 −6
−5
𝑚𝑘 =
=
0 −−2
2
0 −5
−5
𝑚𝑛 =
=
−4 −−6
2
K and n are parallel (they have
the same slope)
J is steepest (it has the
greatest absolute value)
Example - What is the slope of any line perpendicular to the line through
(-2,5) and (3, -1)?
m=
−1 −5
3 −−2
=
−6
5
Perpendicular Slope 
5
6
Example - Given points A(1,-4), B(-1,2),
C(4,2), and D(5,-1), use slopes to determine
whether AC ⊥ DB.
𝑚𝐴𝐶 =
2 −−4
4 −1
6
3
= =2
𝑚𝐷𝐵 =
−1 −2
5 −−1
−3
6
=
=
1
−
2
The slopes are negative
−1
reciprocals (2 * = −1) so
2
the lines are perpendicular.
𝑚1 =
6 −−2
5 −1
8
4
= =2
𝑚𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟
1
=−
2
Example - Line q passes through the points
(1, 2) and (-4, 5). Line t passes through the
points (-2, -1) and (10, 7). Which line is
steeper, q or t?
𝑚𝑞 =
5 −2
−4 −1
𝑚𝑡 =
7 −−1
10 −−2
=
3
−5
=
8
12
=
𝟑
−
𝟓
𝟐
𝟑
= -.6
= = . 𝟔𝟔𝟔𝟕
T is steeper
because it’s
absolute value is
larger.