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).

Answer

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-coordinateof A from the y-coordinateof B.

=6-2=4

The horizontal distance can be found by subtracting the x-coordinateof A from the x-coordinateof B.

=11-3=8

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

Right angled triangle. Hypotenuse A to B unmarked. Other sides 8 and 4.

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.

Right angled triangle. Hypotenuse A to B unmarked. Other sides 7 and 7.

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

 \frac{7}{7}=1

Gradient of line AB = -1 (negative one)