Gradient, length and midpoint

Revise Gradient, length and midpoint for Mathematics D 4024 (O Level) — revision notes, 42 practice questions and instant AI marking.

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Cambridge O Level Mathematics (Syllabus D) 4024

Gradient, length and midpoint

The big idea: Two points fix a line segment. From their coordinates you can find how steep it is (the gradient), how long it is (Pythagoras), and where its middle is (the mean of the coordinates).

Summary — What This Chapter Covers

Find the gradient of a straight line from a graph.
Calculate the gradient from the coordinates of two points on the line.
Calculate the length of a line segment.
Find the coordinates of the midpoint of a line segment.

1. Gradient

Syllabus 3.3 · Paper 1 & 2

Change in y ÷ change in x

The gradient measures steepness: how much y increases when x increases by 1. Lines going up from left to right have a positive gradient; lines going down have a negative gradient. Horizontal lines have gradient 0. From a graph, pick two points far apart on the line, draw a right-angled triangle, and divide the vertical change by the horizontal change.

Gradient of the line through (x₁, y₁) and (x₂, y₂) (not given) m = (y₂ − y₁) ÷ (x₂ − x₁)

Worked Example

Find the gradient of the line through (−2, 5) and (4, −7).

Practice Question 1

The line through (2, k) and (5, 11) has gradient 3. Find k.

2. Length of a Line Segment

Syllabus 3.4 · Paper 1 & 2

Pythagoras on the coordinates

Draw the right-angled triangle whose horizontal side is the change in x and whose vertical side is the change in y. The segment is the hypotenuse.

Length (not given) length = √[(x₂ − x₁)² + (y₂ − y₁)²]

Worked Example

Find the length of the line segment joining (−3, 4) and (2, −8).

Practice Question 2

Find the length of the segment joining (1, 3) and (5, 5). Give your answer as a surd in its simplest form.

3. Midpoint of a Line Segment

Syllabus 3.4 · Paper 1 & 2

The mean of the coordinates

The midpoint is halfway along in both directions, so add the x-coordinates and halve, and add the y-coordinates and halve.

Midpoint (not given) M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

Worked Example

Find the midpoint of the line joining (−3, 8) and (9, −2).

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Gradient, length and midpoint