Gradient

 The gradient or slope of a line is defined as the ratio \( \frac{y-step}{x-step} \)   in going from one point to another on the line.

 Gradient of AB = \( \frac{increase \; in \; y \; from \; A \; to \; B}{increase \; in \; x \; from \; A \; to \; B} \\ = \frac{CB}{AC} \)

Angle of slope

tan θ = \(\frac{opp}{adj} \\ = \frac{AC}{AB} \)

∴ The gradient of a line = tan (angle the line makes with the +ve x-axis)

Gradient of a line joining two points

The gradient of line joining A(x₁, y₁) and B(x₂, y₂)

= \( \frac{y_2 \; – \; y_1}{x_2 \; – \; x_1} \)

or \( \frac{y_1 \; – \; y_2}{x_1 \; – \; x_2} \)

Note that the coordinates must be subtracted in the same order.

Example:

Find the gradient of the line joining (3, -2) and (-5, -8) and the angle it makes with the positive x-axis.

Solution

Equation for Gradient of a line

= \( \frac{y_2 \; – \; y_1}{x_2 \; – \; x_1} \\ = \frac{-8 \; – \; (-2)}{-5 \; – \; (3)} \\ = \frac{-6}{-8} \\ = \frac{3}{4} \)

Angle = tanθ = \(\frac{opp}{adj}\\ = \frac{3}{4} \\ = \scriptsize 0.75 \)

θ = tan^-1(0.75) = 36.87°