Lesson 3: Slope

Property

Given two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ on a line, the slope $m$ of the line is calculated using the formula:

Examples

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:

Property

A point $(x_3, y_3)$ lies on the line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ if the slope between $(x_1, y_1)$ and $(x_3, y_3)$ is the same as the slope of the line.

Examples

• Does the point $(6, 7)$ lie on the line that passes through $(0, 1)$ and $(3, 4)$?

The slope of the line is $m = \frac{4 - 1}{3 - 0} = \frac{3}{3} = 1$.
The slope from $(0, 1)$ to $(6, 7)$ is $m = \frac{7 - 1}{6 - 0} = \frac{6}{6} = 1$.
Since the slopes are equal, the point $(6, 7)$ is on the line.

• Does the point $(2, 5)$ lie on the line that passes through $(-1, -1)$ and $(1, 3)$?

The slope of the line is $m = \frac{3 - (-1)}{1 - (-1)} = \frac{4}{2} = 2$.
The slope from $(-1, -1)$ to $(2, 5)$ is $m = \frac{5 - (-1)}{2 - (-1)} = \frac{6}{3} = 2$.
Since the slopes are equal, the point $(2, 5)$ is on the line.

Explanation

This skill uses the core concept that the slope is constant everywhere on a line. To check if a specific point is on a line, you can calculate the slope of the original line using two given points. Then, calculate the slope between one of the given points and the point you are testing. If the slopes are identical, the test point must lie on the same line.