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medium difficulty5 min

Collinearity using area

Three points are collinear if they lie on one straight line.

In one line: Zero area means one straight line.

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Understand it

The easiest coordinate test for collinearity is the area method. If the area of the triangle formed by the three points is zero, then the points do not make a triangle at all; they lie on the same line. This is a fast and dependable exam check.

Example

The points (1, 1), (3, 3), and (5, 5) have zero area, so they are collinear.

Formula and rule

If 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| = 0, then the points are collinear.

See it worked out

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

Worked example

  1. 1For A(1, 1), B(3, 3), and C(5, 5), the area formula gives 1/2 |1(3 - 5) + 3(5 - 1) + 5(1 - 3)| = 1/2 |-2 + 12 - 10| = 0.
  2. 2So the points are collinear.

Method you can reuse

  1. 1Substitute the three points in the area formula.
  2. 2Simplify carefully.
  3. 3If the result is 0, conclude collinear.
  4. 4If the result is non-zero, the points are not collinear.

Sketch it in your answer

Plot the three points on graph paper and check whether they fall on one straight line, but confirm with the area formula in calculation questions.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What does zero area of a triangle formed by three points tell you?

Think first, then show answer
It tells you that the three points are collinear, because they lie on the same straight line and do not form a real triangle.

Common wrong answer

A common wrong answer is saying the points are only symmetric or equally spaced, which does not replace the zero-area test.

Don't mix this up

Students may rely only on a rough graph or slope idea and miss a calculation error.

Exam tip

If the area becomes zero, stop and write collinear. Do not continue to search for triangle properties.

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

Decide whether three given points are collinear, or show that a triangle cannot be formed from the coordinates.
  1. 1

    Enough for 1 mark

    Three points are collinear if they lie on one straight line.

  2. 2

    Add for 2 marks

    Example: The points (1, 1), (3, 3), and (5, 5) have zero area, so they are collinear.

  3. 3

    Add for 3 marks

    The easiest coordinate test for collinearity is the area method.

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