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Distance formula

The distance formula gives the straight-line distance between two points on the coordinate plane.

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Student-friendly explanation

It comes from the Pythagorean theorem applied to the horizontal and vertical changes between two points. First find the difference in x-coordinates and y-coordinates, then square both changes, add them, and take the square root. This method works for every pair of points, even when they lie in different quadrants.

How to write this in exams

  1. 1

    Start with the exact idea

    The distance formula gives the straight-line distance between two points on the coordinate plane.

  2. 2

    Then show how to use it

    1. Identify x1, y1, x2, y2. 2. Subtract x-coordinates and y-coordinates. 3. Square both differences. 4. Add them. 5. Take the square root and simplify.

  3. 3

    Add one concrete example

    For points A(2, 5) and B(6, 8), the horizontal change is 4 and the vertical change is 3, so the distance is sqrt(4^2 + 3^2) = 5 units.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is 7 units for points (2, 5) and (6, 8), because students add 4 and 3 instead of using Pythagoras.

Definition

The distance formula gives the straight-line distance between two points on the coordinate plane.

Example

For points A(2, 5) and B(6, 8), the horizontal change is 4 and the vertical change is 3, so the distance is sqrt(4^2 + 3^2) = 5 units.

Rule to remember

d = sqrt((x2 - x1)^2 + (y2 - y1)^2). The changes in x and y form the two legs of a right triangle.

Memory hook

Move sideways, move up or down, then use the root.

Examples and method

Worked example

Find the distance between P(-1, 4) and Q(3, 1). Here, x-change = 3 - (-1) = 4 and y-change = 1 - 4 = -3. So distance = sqrt(4^2 + (-3)^2) = sqrt(16 + 9) = 5 units.

Method to apply

1. Identify x1, y1, x2, y2. 2. Subtract x-coordinates and y-coordinates. 3. Square both differences. 4. Add them. 5. Take the square root and simplify.

Diagram support

Draw a right triangle by moving horizontally from one point and then vertically to the other point. The slant side is the distance.

How CBSE asks it

Find the length of a segment joining two given points, or use distances to test whether sides of a triangle are equal.

Avoid common mistakes

Common confusion

Students often add the coordinate differences directly instead of squaring them and taking the square root.

Common wrong answer

A common wrong answer is 7 units for points (2, 5) and (6, 8), because students add 4 and 3 instead of using Pythagoras.

Exam tip

Write the coordinates in the correct order first, then use the formula carefully. The sign will not matter after squaring, but the coordinates must still be read correctly.

Quick check

What is the distance between the points (1, 2) and (4, 6)?

The distance is 5 units because the horizontal change is 3 and the vertical change is 4, so the formula gives sqrt(3^2 + 4^2) = 5.

Answer writing and exam use

1-mark answer

The distance formula gives the straight-line distance between two points on the coordinate plane.

2-mark answer

The distance formula gives the straight-line distance between two points on the coordinate plane. d = sqrt((x2 - x1)^2 + (y2 - y1)^2). The changes in x and y form the two legs of a right triangle. For points A(2, 5) and B(6, 8), the horizontal change is 4 and the vertical change is 3, so the distance is sqrt(4^2 + 3^2) = 5 units.

3-mark answer

It comes from the Pythagorean theorem applied to the horizontal and vertical changes between two points. First find the difference in x-coordinates and y-coordinates, then square both changes, add them, and take the square root. This method works for every pair of points, even when they lie in different quadrants. d = sqrt((x2 - x1)^2 + (y2 - y1)^2). The changes in x and y form the two legs of a right triangle. Find the distance between P(-1, 4) and Q(3, 1). Here, x-change = 3 - (-1) = 4 and y-change = 1 - 4 = -3. So distance = sqrt(4^2 + (-3)^2) = sqrt(16 + 9) = 5 units. Find the length of a segment joining two given points, or use distances to test whether sides of a triangle are equal. A common wrong answer is 7 units for points (2, 5) and (6, 8), because students add 4 and 3 instead of using Pythagoras.
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