# Calculating the gradient using coordinates

Before doing this section you should look at the National 4 Lifeskills Maths section on Gradient of a slope.

The National 4 section shows how to calculate the gradient of a slope using the vertical height and horizontal distance.

A similar approach can be used for calculating the gradient of a line between two known points on a coordinate diagram.

### Example

Calculate the gradient of the line joining point $$A\,\,(3,2)$$ and the point $$B\,\,(11,6)$$.

Plot the points on square paper and you will see that line $$AB$$ is sloping up, therefore the gradient is positive.

The vertical height can be found by subtracting the $$y-coordinate$$of $$A$$ from the $$y-coordinate$$of $$B$$.

$=6-2=4$

The horizontal distance can be found by subtracting the $$x-coordinate$$of $$A$$ from the $$x-coordinate$$of $$B$$.

$=11-3=8$

Make a right-angled triangle with the line $$AB$$ as hypotenuse.

Gradient of line $$AB\, = \frac{{vertical\,height}}{{horizontal\,distance}}$$

$=\frac{4}{8}$

$=\frac{1}{2}$

Now try this question.

Question

Calculate the gradient of the line joining point $$A\,\,(-2,8)$$ and the point $$B\,\,(5,1)$$.

Plot the points on square paper and you will see that line $$AB$$ is sloping down, therefore the gradient is negative.

Make a right angled triangle with the line $$AB$$ as hypotenuse.

Gradient of line $$AB\, = \frac{{vertical\,height}}{{horizontal\,distance}}$$

$\frac{7}{7}=1$

Gradient of line $$AB = -1$$ (negative one)