Similar Triangles and Slope
CC8.EE.6 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; derive the equation y = mx for a line through
the origin and the equation y = mx + b for a line
intercepting the vertical axis at b.
Similar Triangles
• Similar Triangles are triangles who have the
same shape, but not necessarily the same size.
The corresponding angles of similar triangles
are congruent and their corresponding sides
are in PROPORTION. The similar triangles
increase or decrease at a constant rate.
How do I know if two triangles are similar? .
If two triangles are similar, the cross products of their
corresponding sides are equal.
5 3

10 6
10  3  5  6
5 units
30  30
Since the cross products of the
corresponding sides are equal, the
triangles are similar.
10 units
3 units
6 units
Rates of Proportionality in a Triangle? Make a rate of
the legs in each of these right triangles and compare
the results. When making your rate, compare the
vertical leg (rise) to the horizontal leg (run).
What did you notice?
4
The red triangle has a rate of 4 to 8 or 8
4
5
The blue triangle has a rate of 5 to 10 or 10
8
5
10
3
The green triangle has a rate of 3 to 6 or
6
3
6
How many triangles do you see?
Find the ratio of vertical
to horizontal leg of each
triangle. Then simplify to
a fraction. The simplified
fraction should be the
SLOPE of the red line.
3
3
7
7
9
9
The SLOPE of the red line
is 1 because all of the
slope ratios simplify to 1.
Coordinate Plane/Ordered Pairs
The rate of each triangle can be simplified to ½ ! What
do you notice about these triangles and their
hypotenuse in the illustration below?
𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑙𝑒𝑔
𝑅𝑖𝑠𝑒
𝑜𝑟
𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑙𝑒𝑔
𝑅𝑢𝑛
Positive slope
Negative slope
Rises from left
to right
Falls from left to
right
𝑜𝑟
3
2
rise  2

run
3
Zero slope
Undefined slope
Horizontal line
Vertical line
rise 0
 0
run 5
rise 5
  Undefined
run 0
Draw triangles to find the slope of the line.
The slope of the red line is negative
since the triangles are moving
down.
For the smaller triangle, the vertical
change is 2 and the horizontal
change is 3.
For the larger triangle, the vertical
change is 4 and the horizontal
change is 6.
4
The slope for the red line must be
6
or 2 .
3