Lesson 9-3: Use Similar Triangles to Determine Slope

Property

To find the slope of a line from its graph:

1. Locate two points on the graph whose coordinates are integers.
2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
3. Count the rise and the run on the legs of the triangle.
4. Take the ratio of rise to run to find the slope, $m = \frac{\operatorname{rise}}{\operatorname{run}}$.

Examples

• A line passes through $(1, 2)$ and $(4, 8)$. The rise is $8 - 2 = 6$ and the run is $4 - 1 = 3$. The slope is $m = \frac{6}{3} = 2$.
• A line passes through $(0, 6)$ and $(2, 2)$. The rise is $2 - 6 = -4$ and the run is $2 - 0 = 2$. The slope is $m = \frac{-4}{2} = -2$.
• A line on a graph connects points $(-3, 1)$ and $(5, 7)$. The rise is $7-1=6$ and the run is $5-(-3)=8$. The slope is $m = \frac{6}{8} = \frac{3}{4}$.

Explanation

To find slope from a graph, pick two easy-to-read points. Count the vertical distance (rise) and horizontal distance (run) to get from one point to the other. The slope is simply the rise divided by the run.

Property

For any two right triangles formed on a straight line using different pairs of points, the ratio of the vertical side (rise) to the horizontal side (run) is constant.

To prove this, we use the AA (Angle-Angle) Similarity Criterion: If two angles of one triangle are congruent to two angles of another, the triangles are similar ($\sim$). Because the "slope triangles" share a 90° angle and congruent corresponding angles (created by the line crossing the horizontal grid lines), they are similar.

Therefore, their side ratios are equal: