Meaning Of Collinear Points - Vedantu

The concept of collinear points plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Knowing how to identify and prove collinear points helps students work confidently with lines, triangles, and coordinate systems in geometry.

What Is Collinear Points?

Collinear points are a set of three or more points that all lie on the same straight line. This concept is essential in geometry, coordinate geometry, and even in understanding shapes and patterns in real life. For example, when students stand in a straight row during assembly or when buildings line up perfectly in a street, those locations are collinear points.

Key Formula for Collinear Points

Here’s the standard formula to check if three points are collinear using coordinates: \( \text{Area} = \frac{1}{2}|x_{1}(y_{2}-y_{3}) + x_{2}(y_{3}-y_{1}) + x_{3}(y_{1}-y_{2})| \) If the area comes out to be 0, the points are collinear. You can also compare the slopes between pairs of points. If the slopes are equal, all three points are collinear.

Step-by-Step Illustration

1. Suppose you have three points: A(2, 4), B(4, 6), and C(6, 8).
2. Calculate the slope between any two points: Slope of AB = (6 - 4) / (4 - 2) = 2 / 2 = 1 Slope of BC = (8 - 6) / (6 - 4) = 2 / 2 = 1 Slope of AC = (8 - 4) / (6 - 2) = 4 / 4 = 1
3. If all slopes are equal, points A, B, and C are collinear.

Collinear Points Examples in Real Life

• Children standing in a straight line for the school assembly
• Street lights placed along the same side of a road
• Eggs arranged in a row inside a carton
• Dots or points marked directly on a ruler or measuring scale

Frequent Errors and Misunderstandings

• Thinking two points need a test for collinearity (any two points are always collinear)
• Forgetting that area being exactly zero proves collinearity
• Mismatching x and y coordinates in slope or area formulas
• Not checking all pairs of slopes when required

Non-Collinear Points vs. Collinear Points

Aspect	Collinear Points	Non-Collinear Points
Position	All on the same line	Not on the same line
Area of Triangle	Zero	More than zero
Slope Condition	All slopes equal	Slopes not all equal
Example	A, B, C in a row	Forming a triangle

Relation to Other Concepts

Understanding collinear points is important for topics like Coordinate Geometry, area of triangle, and determinant method. Once you know how to check if points are collinear, you can solve questions involving triangles, straight lines, and even section formulas.

Try These Yourself

• Are the points (0, 1), (1, 2), and (2, 3) collinear?
• Find if the area formed by the points (3, 5), (7, 9), and (11, 13) is zero or not.
• Check using distance formula if the points (1, 1), (4, 4), and (7, 7) are collinear.
• Identify which set is non-collinear: {(1,2), (2,3), (4, 6)} or {(0,0),(2,2),(2,5)}

Classroom Tip

To visually check if points are collinear, draw all the points on squared graph paper, then use a ruler to see if they lie exactly on a straight line. Vedantu’s teachers often remind students: “If your pen doesn’t have to jump to touch all points, they are collinear!”

We explored collinear points—from basic meaning, practical formula, step-by-step examples, and links to other geometry topics. Keep practicing with Vedantu to get strong at spotting and proving collinearity, both for exams and for higher-level maths.

Explore More Topics: Coordinate Geometry | Area of Triangle in Coordinate Geometry | Determinant to Find the Area of a Triangle | Section Formula