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What is the Condition for Collinearity of Three Points?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
Three points are said to be collinear if they all lie on the same straight line. The main condition for three points to be collinear is that the area of the triangle formed by these three points must be zero.
Simple Example
Quick Example
Imagine three friends, Rohan, Priya, and Amit, standing in a straight line for a school assembly. If you were to draw an imaginary line connecting them, all three would fall on that single line. This means their positions are 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: Recall the formula for the area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3): Area = 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
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Step 2: Assign the coordinates: x1=1, y1=2; x2=3, y2=4; x3=5, y3=6.
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Step 3: Substitute the values into the formula:
Area = 1/2 |1(4 - 6) + 3(6 - 2) + 5(2 - 4)|
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Step 4: Calculate the terms inside the absolute value:
Area = 1/2 |1(-2) + 3(4) + 5(-2)|
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Step 5: Simplify the terms:
Area = 1/2 |-2 + 12 - 10|
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Step 6: Further simplify:
Area = 1/2 |0|
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Step 7: Calculate the final area:
Area = 0
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Answer: Since the area of the triangle formed by points A, B, and C is 0, the points A(1, 2), B(3, 4), and C(5, 6) are collinear.
Why It Matters
Understanding collinearity is crucial in fields like Engineering and AI/ML for tasks such as path planning for robots or recognizing patterns in data. For example, city planners use it to design straight roads efficiently, and game developers use it for character movement.
Common Mistakes
MISTAKE: Forgetting the 1/2 in the area formula or making sign errors when substituting coordinates. | CORRECTION: Always double-check your formula and calculations, especially the subtraction within the parentheses.
MISTAKE: Assuming points are collinear if only two points are given. | CORRECTION: Collinearity is a property of three or more points. Two points always define a straight line, so they are always collinear by definition.
MISTAKE: Not understanding that the absolute value in the area formula means the result must be exactly zero, not just a very small number. | CORRECTION: For collinearity, the calculated value inside the absolute sign must simplify to exactly zero.
Practice Questions
Try It Yourself
QUESTION: Are the points P(0, 0), Q(2, 2), and R(4, 4) collinear? | ANSWER: Yes
QUESTION: Find the value of 'k' if the points A(1, 5), B(2, 6), and C(k, 7) are collinear. | ANSWER: k = 3
QUESTION: The vertices of a triangle are (x, 0), (0, y) and (1, 1). If the points are collinear, prove that 1/x + 1/y = 1. | ANSWER: (This is a proof, so no single numerical answer. The student needs to show the steps using the area = 0 condition.)
MCQ
Quick Quiz
Which of the following conditions ensures that three points are collinear?
The sum of their x-coordinates is zero.
The area of the triangle formed by them is zero.
They are equally spaced from each other.
Their y-coordinates are all positive.
The Correct Answer Is:
B
For three points to be collinear, they must lie on a single straight line. This means they cannot form a 'proper' triangle, so the area of the triangle they form must be zero. Options A, C, and D are not general conditions for collinearity.
Real World Connection
In the Real World
Think about how self-driving cars (like those being tested in some Indian cities) navigate. They use sensors to map points in their surroundings. To drive in a straight line, the car's computer needs to constantly check if the points it aims for are collinear, ensuring a smooth and safe path without swerving.
Key Vocabulary
Key Terms
COLLINEAR: Points lying on the same straight line | COORDINATES: A set of values that show an exact position on a graph | AREA OF TRIANGLE: The space enclosed by a triangle, calculated using a specific formula | VERTICES: The corner points of a shape | ABSOLUTE VALUE: The non-negative value of a number, ignoring its sign.
What's Next
What to Learn Next
Next, you can explore how to find the equation of a straight line passing through two given points. This builds on collinearity because all points on that line are collinear, and you'll use similar coordinate geometry concepts.
