medium difficulty5 min

Distance Between Two Points

The distance between two points (x1, y1) and (x2, y2) is the length of the straight line segment joining them on the Cartesian plane.

In one line: Difference, square, add, root.

Practice This Concept

Understand it

The distance formula comes from Pythagoras theorem. The horizontal difference is x2 - x1 and the vertical difference is y2 - y1. These act like the two perpendicular sides of a right triangle, and the required distance is the hypotenuse.

Example

Distance between (1, 2) and (4, 6) is √((4 - 1)² + (6 - 2)²) = √(9 + 16) = 5 units.

Formula and rule

Distance formula: d = √((x2 - x1)² + (y2 - y1)²).

It is derived using Pythagoras theorem.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

For A(-2, 1) and B(4, 9), d = √((4 + 2)² + (9 - 1)²) = √(36 + 64) = 10 units.

Method you can reuse

Write both points, find x-difference, find y-difference, square both differences, add them, then take the square root.

Visual tip

A right triangle can be drawn by dropping horizontal and vertical segments between the two points before applying Pythagoras theorem.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Find the distance between (0, 0) and (3, 4).

Think first, then show answer

The distance is √((3 - 0)² + (4 - 0)²) = √(9 + 16) = sqrt25 = 5 units.

Common wrong answer

For (-2, 1) and (4, 9), writing x-difference as 4 - 2 instead of 4 - (-2) gives 2 and leads to a wrong distance.

Don't mix this up

Students sometimes add coordinates directly instead of subtracting corresponding coordinates.

Exam tip

Always subtract x-coordinates together and y-coordinates together; the squares make the final distance non-negative.