1. Slope and Similar Triangles
Objective: In this lesson you will learn to derive slope by using similar triangles.

2. Congruent Triangles What are congruent triangles?
Congruent triangles are triangles that have the same size and the same shape.
The corresponding side lengths of congruent triangles are equal.

3. Congruent Triangles
Vertical
Length
Horizontal
Length

4. Slope The SLOPE of this line is
Note that the ratio of the vertical side length to the horizontal side length of each triangle is also

5. Slope
What is Slope?
Slope is the rate of change between any two points.
Reads “rise over run”

6. More Slope The slope of a line can sometimes be positive or negative
The side lengths of each triangle formed are always positive
When the slope is negative, the ratio of the side lengths of the triangles remains positive
Note that the ratio of the side lengths is positive. However, the slope is negative.

7. Try This!!!! Write the slope of the line.
Horizontal length 3
Vertical length 2
Note that the ratio of the side lengths is positive.

8. Try This!!!! Write the slope of the line.
Horizontal length 3
Vertical length 4
Note that the ratio of the side lengths is positive. However, the slope is negative.

9. Similar Triangles What are Similar Triangles?
Similar triangles are triangles that have the same shape but not necessarily the same size.
Notice that the angles are congruent (the same measure), however the side lengths are different.

10. Slopes and Similar Triangles
Simplest form

11. Slope and Similar Triangles
The corresponding side lengths of similar triangles are proportional.
Recall that two quantities are proportional if they have the same ratio.
The side lengths of the similar triangles have the same simplified ratio
Smaller slope
Larger slope

12. Try This!!!! Find the simplified slopes of similar triangles.
Smaller slope
Larger slope

13. Example 1:
Find the slope.

14. Example 2:
Find the slope.

15. Example 3
Draw the graph of a line with a positive slope. Then draw the triangles formed by the slope of the line Show that the simplified ratio of each triangle is equivalent to the slope.