Year 10/11 maths: Using gradients to find equations of lines

On a graph, parallel lines have the same gradient.

For example, y = 3x + 3 and y = 4 + 3x are parallel because they both have a gradient of 3.

The gradient is the value before the x. The other value is where the graph **C**rosses the y-axis

Q. What is the equations of the lines shown.

ANSPerpendicular lines cross at right angles.

In the diagram the lines are perpendicular.

The gradients of these two lines are ½ and –2

Multiply line gradients to test if they are perpendicular.

If the answer is –1 they are perpendicular.

In this case ½ × –2 = –1

So *flip* the first gradient and change the sign

e.g. 1 and –1 are perpendicular

Q1. Find the equation of the perpendicular line to the line shown, which passes through the point (0, 2).

i.e. Work out the gradient and flip and change the sign to get the perpendicular gradient first

Q2. Find equation of perpendicular line to

y = 4 – ¾x which passes through (0, –3)

ANS

To find the gradient of a line draw a right angled triangle, then using the 2 points work out the (difference in y)/(difference in x)

e.g. Find the gradient of the line from (1, 2) to (4, 8)

The difference in y = 8 – 2 = 6

The difference in x = 4 – 1 = 3

Gradient = 6/3 = 2 . Note it is positive.

**Q1.** Work out the gradient between the two points on the diagram.

(Hint: first find the two co-ordinates,
then work out the difference (△) in x and y and divide the two.

**Q2.** Work out the equation of the line passing through these points

Give your answer in the form ax + by + c = 0

Find equation of perpendicular line. Menu

Point A has the coordinates (7,2) and point B has the coordinates (1,6)

Q1. Find the equation of a line AB.

Give your answer in the form ax + by + c = 0

Q2. Find the equation of a line that is perpendicular to AB and passes through (0, 0).

Give your answer in the form ax + by + c = 0

ANS

A tangent just touches a circle.

We can use a circle theorem to help us find the equation of a tangent.

Circle theorems show that the angle between a tangent and a radius in a circle is 90°

Q. A circle has centre (6, 8).

A tangent touches the circle at (3, 12)

Find the equation of the tangent

ANS