What is the midpoint of (2, 6) and (8, 10)?

(5, 8). Average the x-coordinates and the y-coordinates.

Which line is parallel to y = 4x − 1?

y = 4x + 7. Parallel lines have the same gradient.

Straight lines, midpoints and perpendicular bisectors · Cambridge IGCSE Additional Mathematics

In plain words

Given two points, you can find everything about the line through them: its slope, its length, its middle, and the line that cuts it in half at right angles. Each is one short formula.

6 things to know

Gradient of the line through (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁) ÷ (x₂ − x₁).
Equation of a line with gradient m through (x₁, y₁): y − y₁ = m(x − x₁).
Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to −1, so one is the negative reciprocal of the other.
Midpoint: the average of the x-coordinates and the average of the y-coordinates.
Length: √((x₂ − x₁)² + (y₂ − y₁)²).
The perpendicular bisector of AB passes through the midpoint of AB and is perpendicular to AB.

Worked example

A is (1, 2) and B is (7, 10). Find the length of AB and the equation of its perpendicular bisector.

Length = √(6² + 8²) = √100 = 10.
Gradient of AB = 8/6 = 4/3, so the perpendicular gradient is −3/4. The midpoint is (4, 6).
Perpendicular bisector: y − 6 = −(3/4)(x − 4).
Multiply by 4: 4y − 24 = −3x + 12, so 3x + 4y = 36.

Tips and tricks

For a perpendicular gradient, turn the fraction upside down and change its sign: 4/3 becomes −3/4.
Subtract in the same order top and bottom when finding a gradient. Mixing the order changes the sign.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Straight lines, midpoints and perpendicular bisectors: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

What is the gradient of the line through (1, 3) and (5, 11)?
- 1/2
- 2 (the answer)
- 4
- 8
(11 − 3) ÷ (5 − 1).
A line has gradient 3. What is the gradient of a line perpendicular to it?
- 3
- −3
- 1/3
- −1/3 (the answer)
3 × (−1/3) = −1.
What is the midpoint of (2, 6) and (8, 10)?
- (6, 4)
- (5, 8) (the answer)
- (10, 16)
- (3, 2)
Average the x-coordinates and the y-coordinates.
What is the distance between (0, 0) and (5, 12)?
- 7
- 12
- 13 (the answer)
- 17
√(25 + 144) = √169.
Which line is parallel to y = 4x − 1?
- y = −4x + 2
- y = 4x + 7 (the answer)
- y = (1/4)x
- y = −(1/4)x + 3
Parallel lines have the same gradient.

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

3 questions, 9 marks. Write your answers on paper, then check them.