GCSE Maths revision · Higher: number and algebra

Perpendicular lines

Higher only

The idea

Perpendicular lines meet at right angles, and their gradients multiply to $-1$ . If one line has gradient $m$ , a perpendicular line has gradient $-\frac{1}{m}$ .

Find the gradient of the given line.
Flip it and change the sign: $\frac{2}{3}$ becomes $-\frac{3}{2}$ .
Use a point on the new line to find $c$ .

Watch out Flip the gradient and change its sign: both steps are needed.

See it worked

Worked exampleNon-calculator3 marks

Line $L$ has equation $y = 2x - 3$ .

(a) Write down the gradient of a line perpendicular to $L$ .

(b) Find the equation of the line perpendicular to $L$ that passes through $(4, 1)$ .

(a) Perpendicular gradients multiply to $-1$ : $2 \times m = -1$ , so $m = -\frac{1}{2}$ .
(b) Substitute $(4, 1)$ into $y = -\frac{1}{2}x + c$ . $1 = -2 + c \Rightarrow c = 3$
$y = -\frac{1}{2}x + 3$ .

Answer (a) $-\frac{1}{2}$ ; (b) $y = -\frac{1}{2}x + 3$

Your turn

Question 1Non-calculator2 marks

Line L has equation y = (2/5)x + 7. Find the equation of the line perpendicular to L that passes through (4, $-1$ ).

Question 2Non-calculator5 marks

A is $(0,0)$ and B is $(10,8)$ .

(a) Find the perpendicular bisector of AB.

(b) Find where the perpendicular bisector meets the x-axis.

Question 3Non-calculator5 marks

A circle has centre O(0, 0). The point P(3, 4) lies on the circle.

(a) Find the gradient of the radius OP.

(b) Find the equation of the tangent to the circle at P.

In past papers

38 official questions in 38 papers since 2017. The latest:

All GCSE Maths past papers

GCSE Maths Pro

£119 once

At least 12 months of access

By paying you agree to our Terms and get access straight away, which ends your 14-day cancellation right.

Start free

No VAT is charged.