Slope of Straight Line joining Points in Cartesian Plane

Theorem

Let $p_1 := \tuple {x_1, y_1}$ and $p_2 := \tuple {x_2, y_2}$ be points in a cartesian plane.

Let $\LL$ be the straight line passing through $p_1$ and $p_2$.

Then the slope of $\LL$ is given by:

$\tan \theta = \dfrac {y_2 - y_1} {x_2 - x_1}$

where $\theta$ is the angle made by $\LL$ with the $x$-axis.

Proof

The slope of a straight line is defined as the change in $y$ divided by the change in $x$.

The change in $y$ from $p_1$ to $p_2$ is $y_2 - y_1$.

The change in $x$ from $p_1$ to $p_2$ is $x_2 - x_1$.

By definition of tangent of $\theta$:

$\tan \theta = \dfrac {y_2 - y_1} {x_2 - x_1}$

Hence the result.

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {III}$. Analytical Geometry: The Straight Line
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 10$: Formulas from Plane Analytic Geometry: $10.2$: Slope $m$ of Line joining Two Points $\map {P_1} {x_1, y_1}$ and $\map {P_2} {x_2, y_2}$