#### Key Concept

You can use similar triangles to explain why the slope *m* is the same between any two distinct points on a non-vertical line in the coordinate plane.

If you begin with a line passing through the origin and the point (*a*, *b*), the ratio of the rise to the run of the triangle is *b* to *a*.

rise
run
=
b
a
fraction rise , over run end fraction . equals , b over eh

Now dilate the triangle by scale factor *k* with center at the origin. The ratio of the rise to the run for the dilation is still *b* to *a*.

rise
run
=
k
b
k
a
=
b
a
table with 2 rows and 2 columns , row1 column 1 , fraction rise , over run end fraction , column 2 equals . fraction cross out enclosing , k , end crossout b , over cross out enclosing , k , end crossout eh end fraction , row2 column 1 , , column 2 equals , b over eh , end table

The ratio is the same as for the original triangle.

You can also show that the ratio of the rise to the run is the same as the slope of the line through the point (*a*, *b*) and (*ka*, *kb*). Use the slope formula.

rise
run
=
b
a
Slope
m
=
change in
y
change in
x
=
k
b
−
b
k
a
−
a
=
b
(
k
−
1
)
a
(
k
−
1
)
=
b
a
table with 5 rows and 1 column , row1 column 1 , fraction rise , over run end fraction . equals , b over eh , row2 column 1 , cap slope , m , column 2 equals . fraction changein . y , over changein . x end fraction , row3 column 1 , , column 2 equals . fraction k b minus b , over k eh minus eh end fraction , row4 column 1 , , column 2 equals . fraction b . cross out enclosing , open k minus 1 close , end crossout , over eh . cross out enclosing , open k minus 1 close , end crossout end fraction , row5 column 1 , , column 2 equals , b over eh , end table

This also works for a non-vertical line that does not pass through the origin. A translation up results in triangles that are congruent to those formed by the line passing through the origin. So the ratio of the rise to the run between any two points on this new line is still *b* to *a*.

rise
run
=
b
a
fraction rise , over run end fraction . equals , b over eh