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medium difficulty8 min

Distance formula

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

In one line: Move sideways, move up or down, then use the root.

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Understand it

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.

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 √(42 + 32) = 5 units.

Formula and rule

d = √((x2 - x1)2 + (y2 - y1)2).

The changes in x and y form the two legs of a right triangle.

See it worked out

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

Worked example

Find the distance between P(-1, 4) and Q(3, 1).

  1. 1Here, x-change = 3 - (-1) = 4 and y-change = 1 - 4 = -3.
  2. 2So distance = √(42 + (-3)2) = √(16 + 9) = 5 units.

Method you can reuse

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

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer
The distance is 5 units because the horizontal change is 3 and the vertical change is 4, so the formula gives √(32 + 42) = 5.

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.

Don't mix this up

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

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.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

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.
  1. 1

    Enough for 1 mark

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

  2. 2

    Add for 2 marks

    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 √(42 + 32) = 5 units.

  3. 3

    Add for 3 marks

    It comes from the Pythagorean theorem applied to the horizontal and vertical changes between two points.

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