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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete 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.

Definition

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

Example

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

Rule to remember

Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2). It is derived using Pythagoras theorem.

Memory hook

Difference, square, add, root.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

How CBSE asks it

Students may be asked to calculate distance, compare side lengths, or prove that two segments are equal.

Avoid common mistakes

Common confusion

Students sometimes add coordinates directly instead of subtracting corresponding coordinates.

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.

Exam tip

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

Quick check

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

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

Study the distance between two points diagram carefully

Use the labelled diagram to keep distance between two points clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of distance between two points in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on distance between two points.

Revision cue

Revise distance between two points through the labels before writing the answer.

Answer writing and exam use

1-mark answer

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

2-mark answer

The distance between two points (x1, y1) and (x2, y2) is the length of the straight line segment joining them on the Cartesian plane. Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2). It is derived using Pythagoras theorem. Distance between (1, 2) and (4, 6) is sqrt((4 - 1)^2 + (6 - 2)^2) = sqrt(9 + 16) = 5 units.

3-mark answer

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. Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2). It is derived using Pythagoras theorem. For A(-2, 1) and B(4, 9), d = sqrt((4 + 2)^2 + (9 - 1)^2) = sqrt(36 + 64) = 10 units. Students may be asked to calculate distance, compare side lengths, or prove that two segments are equal. For (-2, 1) and (4, 9), writing x-difference as 4 - 2 instead of 4 - (-2) gives 2 and leads to a wrong distance.
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