Length and Midpoint

2026 Syllabus Objectives

By the end of this topic, you should be able to:

Calculate the length of a line segment
Find the coordinates of the midpoint of a line segment

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Understanding Coordinates

Before we begin, let's make sure we understand coordinates. A coordinate is a pair of numbers that tells us the exact position of a point on a grid (called a coordinate plane). We write coordinates in brackets like this: (x, y).

The x-coordinate tells us how far left or right the point is
The y-coordinate tells us how far up or down the point is

For example, the point A(3, 4) means:

Move 3 units along the horizontal axis (the x-axis)
Then move 4 units up the vertical axis (the y-axis)

When we have two points, like A(3, 4) and B(5, 8), we can join them with a straight line. This straight line is called a line segment.

1. Calculating the Length of a Line Segment (Distance Formula)

The length of a line segment is simply how long the line is between two points. We also call this the distance between the two points.

To find this distance, we use a special formula called the Distance Formula.

The Distance Formula

If we have two points:

Point A with coordinates (x₁, y₁)
Point B with coordinates (x₂, y₂)

The distance between them is:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

This formula comes from something you might have learned called Pythagoras' theorem (the rule about right-angled triangles).

How to Use the Distance Formula (Step-by-Step)

Let's break this down into simple steps:

Step 1: Write down the coordinates of both points clearly

Label which is (x₁, y₁) and which is (x₂, y₂)

Step 2: Find the difference in x-coordinates

Calculate: x₂ - x₁

Step 3: Find the difference in y-coordinates

Calculate: y₂ - y₁

Step 4: Square both differences

Calculate: (x₂ - x₁)²
Calculate: (y₂ - y₁)²

Step 5: Add the two squared values together

Step 6: Find the square root of your answer

Worked Example 1: Finding Distance

Question: Find the distance between point A(3, 4) and point B(5, 8).

Solution:

Step 1: Identify the coordinates

Point A: x₁ = 3, y₁ = 4
Point B: x₂ = 5, y₂ = 8

Step 2: Find the difference in x-coordinates

x₂ - x₁ = 5 - 3 = 2

Step 3: Find the difference in y-coordinates

y₂ - y₁ = 8 - 4 = 4

Step 4: Square both differences

(x₂ - x₁)² = 2² = 4
(y₂ - y₁)² = 4² = 16

Step 5: Add them together

4 + 16 = 20

Step 6: Find the square root

Distance = √20 = 4.472 (to 3 decimal places)

Answer: The distance between A and B is √20 or approximately 4.472 units.

Worked Example 2: Finding Distance with Negative Coordinates

Question: Find the distance between point C(-2, 1) and point D(3, -4).