How to Prove Three Points are Collinear?
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
- Suppose you have three points: A(2, 4), B(4, 6), and C(6, 8).
- Calculate the slope between any two points:
Slope of AB = (6 - 4) / (4 - 2) = 2 / 2 = 1Slope of BC = (8 - 6) / (6 - 4) = 2 / 2 = 1Slope of AC = (8 - 4) / (6 - 2) = 4 / 4 = 1 - 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
FAQs on Collinear Points – Meaning, Formula & Examples
1. What are collinear points in geometry?
In geometry, collinear points are three or more points that lie on the same straight line. This means you can draw a single straight line that passes through all of them.
2. How do you prove three points are collinear using coordinates?
There are several ways to prove collinearity using coordinates: * **Method 1: Slope Formula:** Calculate the slope between each pair of points. If the slopes are equal for all pairs, the points are collinear. * **Method 2: Area of a Triangle Formula:** Calculate the area of the triangle formed by the three points using the determinant formula. If the area is zero, the points are collinear. * **Method 3: Distance Formula:** Calculate the distance between each pair of points. If the sum of the distances between two pairs of points equals the distance between the remaining pair, the points are collinear.
3. What is the formula for checking collinearity using the area of a triangle?
For points A(x₁,y₁), B(x₂,y₂), and C(x₃,y₃), the area of triangle ABC is given by: Area = ½ |x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)| . If the area is 0, the points are collinear.
4. Can you have only two collinear points?
Yes, any two points are always collinear because you can always draw a straight line through them. However, the term 'collinear' is typically used to describe three or more points.
5. What are non-collinear points?
Non-collinear points are points that do not all lie on the same straight line. You cannot draw a single straight line that passes through all of them.
6. Why is collinearity important in triangles and polygons?
Collinearity is crucial in geometry because: * It helps determine the area of a triangle (if points are collinear, the area is zero). * It's used to prove geometric properties and theorems relating to lines and concurrency. * It simplifies calculations and problem-solving in various geometric contexts.
7. Is the area method always better than the slope method for checking collinearity?
No, both the area and slope methods are valid. The area method is generally more versatile, working well with any set of three points. The slope method, however, requires that the x-coordinates of the points are distinct. Choose the method best suited to the problem.
8. How are collinear points used in real-world applications?
Collinearity has practical applications in: * Surveying and mapping * Computer graphics and image processing * Civil engineering and city planning * Alignment and precision tasks
9. What is the difference between collinear and coplanar points?
Collinear points lie on the same line. Coplanar points lie on the same plane (a flat surface). All collinear points are coplanar, but not all coplanar points are collinear.
10. How do you explain collinear points to a younger student?
Imagine several dots on a piece of paper. If you can draw a straight line through *all* the dots without lifting your pencil, those dots are collinear.
11. Can collinearity be used in higher mathematics?
Yes, the concept of collinearity extends to advanced mathematical concepts such as vector geometry, coordinate geometry, and analytic proofs. It is a fundamental idea that underlies many more complex geometric relationships.
