What is Collinearity of Three Points?

S3-SA1-0377

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Collinearity of three points means that these three points lie on the same straight line. Imagine drawing a single straight line; if all three points can be placed perfectly on that line, they are collinear.

Simple Example

Quick Example

Think about three friends, Rohan, Priya, and Sameer, standing in a queue for a movie ticket. If they are standing one behind the other in a perfectly straight line, then their positions are collinear. If Priya steps out of line, they are no longer collinear.

Worked Example

Step-by-Step

Let's check if points A(1, 2), B(3, 4), and C(5, 6) are collinear.

Step 1: Calculate the slope of the line segment AB.
Slope of AB = (y2 - y1) / (x2 - x1) = (4 - 2) / (3 - 1) = 2 / 2 = 1

---
Step 2: Calculate the slope of the line segment BC.
Slope of BC = (y2 - y1) / (x2 - x1) = (6 - 4) / (5 - 3) = 2 / 2 = 1

---
Step 3: Compare the slopes. If the slopes of AB and BC are equal, and point B is common to both segments, then the points A, B, and C are collinear.

---
Answer: Since the slope of AB is 1 and the slope of BC is 1, and B is a common point, points A, B, and C are collinear.

Why It Matters

Understanding collinearity helps in fields like computer graphics to draw straight lines or in physics to analyze forces acting along a single path. Engineers use it when designing structures to ensure parts align correctly, and data scientists use it to understand linear relationships in data.

Common Mistakes

MISTAKE: Only checking the slope of AB and AC, without ensuring a common point. | CORRECTION: To confirm collinearity, check the slope of AB and BC (or AC and BC). The middle point must be common to both segments whose slopes you are comparing.

MISTAKE: Confusing collinearity with points being in the same plane. | CORRECTION: Collinearity specifically means points are on the SAME STRAIGHT LINE. Points can be in the same plane but not on the same line (e.g., vertices of a triangle).

MISTAKE: Incorrectly applying the distance formula instead of the slope formula for collinearity. | CORRECTION: While the distance formula can also be used (if AB + BC = AC), the slope formula (slope of AB = slope of BC) is often simpler and more direct for checking collinearity.

Practice Questions

Try It Yourself

QUESTION: Are the points P(0, 0), Q(2, 2), and R(4, 4) collinear? | ANSWER: Yes, they are collinear.

QUESTION: Determine if points X(1, 5), Y(3, 7), and Z(5, 10) are collinear. | ANSWER: No, they are not collinear.

QUESTION: Find the value of 'a' if the points A(1, 3), B(2, a), and C(3, 7) are collinear. | ANSWER: a = 5

MCQ

Quick Quiz

Which of the following conditions must be true for three points A, B, and C to be collinear?

The distance AB is equal to the distance BC.

The slope of AB is equal to the slope of BC.

All three points form an equilateral triangle.

The points are in the same plane.

The Correct Answer Is:

For three points to be collinear, they must lie on the same straight line, which means the slope between any two pairs of points (like AB and BC) must be the same. Option A is not always true, and C and D do not guarantee collinearity.

Real World Connection

In the Real World

Imagine a drone delivering packages for a service like Zepto. If the drone needs to fly directly from point A to B, and then B to C, for maximum efficiency, points A, B, and C should be collinear. This helps in path planning and fuel optimization for delivery services.

Key Vocabulary

Key Terms

SLOPE: A measure of the steepness or gradient of a line.| COORDINATES: A set of numbers that indicate the position of a point on a graph.| LINE SEGMENT: A part of a line that is bounded by two distinct endpoints.| STRAIGHT LINE: A one-dimensional figure that has no thickness and extends infinitely in both directions.

What's Next

What to Learn Next

Now that you understand collinearity, you can explore concepts like finding the equation of a straight line or calculating the area of a triangle using coordinates. These build directly on understanding how points relate to lines.