How to check if three points are collinear using slope formula and area method
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 in Geometry Explained Clearly
1. What are collinear points in Maths?
Collinear points are three or more points that lie on the same straight line. In coordinate geometry, if multiple points share a single straight line, they are called collinear points.
- If points A, B, and C lie on one straight line, they are collinear.
- If even one point does not lie on that line, the points are non-collinear.
- Collinearity is commonly checked using slope or area methods.
2. How do you check if three points are collinear?
Three points are collinear if the slopes between each pair of points are equal.
- Step 1: Find slope of AB: m₁ = (y₂ − y₁)/(x₂ − x₁)
- Step 2: Find slope of BC: m₂ = (y₃ − y₂)/(x₃ − x₂)
- Step 3: If m₁ = m₂, the points are collinear.
3. What is the formula for collinearity of three points?
Three points are collinear if the area of the triangle formed by them is 0. The formula is:
Area = ½ | x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) |.
- If Area = 0, the points are collinear.
- If Area ≠ 0, they form a triangle.
4. Can you give an example of collinear points?
Yes, the points (1,2), (2,4), and (3,6) are collinear points.
- Slope between (1,2) and (2,4) = (4−2)/(2−1) = 2
- Slope between (2,4) and (3,6) = (6−4)/(3−2) = 2
- Since the slopes are equal, the three points lie on the same straight line.
5. What is the difference between collinear and non-collinear points?
Collinear points lie on the same straight line, while non-collinear points do not.
- Collinear points: Form a straight line and have equal slopes.
- Non-collinear points: Do not lie on one line and form a triangle.
- Area test: 0 for collinear, non-zero for non-collinear.
6. Why is the area of a triangle zero for collinear points?
The area of a triangle is zero for collinear points because they do not form an actual triangle.
- A triangle requires three non-collinear points.
- If all three points lie on one straight line, the height becomes zero.
- Since Area = ½ × base × height, if height = 0, then Area = 0.
7. How do you prove points are collinear using slopes?
Points are proven collinear if the slopes between pairs of points are equal.
- Calculate slope AB.
- Calculate slope BC.
- If slope AB = slope BC, then A, B, and C are collinear.
8. Are two points always collinear?
Yes, any two points are always collinear because exactly one straight line passes through them.
- Collinearity is meaningful when checking three or more points.
- With only two points, a unique line always exists.
9. How do you check collinearity using determinants?
Three points are collinear if the determinant of their coordinate matrix is zero.
Set up:
| x₁ y₁ 1 |
| x₂ y₂ 1 |
| x₃ y₃ 1 |
- If the determinant = 0, the points are collinear.
- If not, they are non-collinear.
10. What are some real-life examples of collinear points?
Collinear points appear in real life whenever objects lie on a single straight path.
- Streetlights placed along a straight road.
- Poles aligned along a railway track.
- Points marked on a ruler.
