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Collinearity using area

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

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

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

Definition

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

Example

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

Rule to remember

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

Memory hook

Zero area means one straight line.

Examples and method

Worked example

For 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. So the points are collinear.

Method to apply

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

Diagram support

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.

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

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.

Exam tip

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

Quick check

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

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

Answer writing and exam use

1-mark answer

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

2-mark answer

Three points are collinear if they lie on one straight line. If 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| = 0, then the points are collinear. The points (1, 1), (3, 3), and (5, 5) have zero area, so they are collinear.

3-mark answer

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. If 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| = 0, then the points are collinear. For 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. So the points are collinear. Decide whether three given points are collinear, or show that a triangle cannot be formed from the coordinates. A common wrong answer is saying the points are only symmetric or equally spaced, which does not replace the zero-area test.
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