Collinear Points - Definition, Formula, Examples

What are Collinear Points?

The word collinear comes from “co” meaning together and “linear” meaning line. So, collinear points are points that lie together in one straight line.

When you have only two points, they are always collinear because you can always draw one straight line through them. The concept becomes more interesting with three or more points. Then you must check whether all of them lie on the same line.

 Read More: Coincident Lines

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 at once.

If you take three points and they form a triangle (that is, a closed shape with area), those points are non-collinear. If they form a straight line with zero area, they are collinear.

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Collinear Points Geometry 

In geometry, collinear points help us understand how points relate to lines and shapes.

• A straight line is determined by any two points. A third point may or may not lie on that same line.

• If it does, the three points are collinear.

• If it does not, they are non-collinear.

Collinear points geometry is used in constructing lines, verifying alignments, and solving geometric problems. It also plays a key role in coordinate geometry, where we work with points on a graph or Cartesian plane.

In higher mathematics, the idea of collinearity is used in theorems about concurrency, midpoints, and triangle centers. Understanding this concept is the first step toward mastering many geometric proofs.

 Read More: Cartesian Plane

Collinear Points Formula

When points are given in coordinate form, we can use a few formulas to check whether they are collinear. There are three main methods: the slope method, the distance method, and the area method.

1. Slope Method

If the slope between the first and second points is the same as the slope between the second and third points, they are collinear.

If A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) are three points, then:

Slope of AB = y2− y₁/ x₂ - x₁

Slope of BC = y3− y2 / x3 - x2

If these two slopes are equal, A, B, and C are collinear.

2. Distance Method

If three points are collinear and the point B lies between A and C, then:

AB + BC = AC

Using the distance formula:

AB = √ (x₂ - x₁)2  ( y₂ - y₁)2

BC = √ (x3​- x₂)2  (y3​- y₂ )2

AC = √ (x3​- x₁ )2  (y3​-  y₁)2

If AB + BC = AC,  then A, B, and C are collinear.

3. Area or Determinant Method

This method uses the area of a triangle formed by three points. If the area is zero, the points are collinear.

Area = 1​/2 ∣ x1​(y2​− y3​) + x2​(y3​− y1​) + x3​(y1​− y2​)∣

If the area equals zero, the points are collinear. This formula is also known as the collinear points formula because it works even when slopes are undefined.

Applications of Collinear Points in Geometry

Understanding collinear points geometry is important because:

1. Simplifies proofs: Many geometry theorems rely on showing that points lie in a line (collinear) or not.

2. Triangles and polygons: If three vertices are collinear, the “triangle” is degenerate (zero area).

3. Constructions: In compass-and-straightedge constructions, sometimes you must ensure some points lie on a line.

4. Coordinates and vectors: In analytic geometry and vector geometry, collinearity is tied to linear combinations and vector relations.

5. Advanced geometry: Projective geometry, concurrency, and collineation maps preserve collinearity.

Also, many contest problems or exam questions ask you to prove whether certain points are collinear or not, using slope, area, or other strategies.

Read More: Before Number Concept

Collinear Points Examples

Let us look at some simple collinear points examples to understand the concept better.

Example 1: Show that the points A(2, 3), B(4, 5), and C(6, 7) are collinear using the Distance Method.

Solution: We will use the Distance Method to test if the points are collinear. According to this method, if the sum of the smaller two distances equals the largest distance, the points lie on the same straight line.

Step 1: Write down the given points
A(2, 3), B(4, 5), C(6, 7)

Step 2: Use the distance formula

Distance =√ (x₂ - x1) + (y₂ - x1)

Step 3: Calculate AB, BC, and AC

AB = √ (4 - 2)2 + (5 - 3)2 =  √ (2)2 + (2)  = √8  = 2√2

BC = √ (6 - 4)2 + (7 - 5)2 =  √ (2)2 + (2)  = √8  = 2√2

AC = √ (6 - 2)2 + (7 - 3)2 =  √ (4)2 + (4)  = √32  = 4√2

Step 4: Check if AB + BC = AC

AB + BC =  2√2 + 2√2 = 4√2 = AC

Since AB + BC= AC, the points A(2, 3), B(4, 5), and C(6, 7) are collinear points.

Example 2: Show that the points P(1, 2), Q(3, 6), and R(5, 10) are collinear using the Slope Method.

Solution: In the Slope Method, if the slope of PQ equals the slope of QR, the points are collinear.

Step 1: Write down the given points

P(1, 2), Q(3, 6), R(5, 10)

Step 2: Formula for slope

m= y2− y₁/ x₂ - x₁

Step 3: Find slope of PQ and QR

Slope of PQ = 6 − 2/ 3 − 1= 4/2 = 2

Slope of QR = 10 − 6/ 5 − 3 = 4/2 = 2

Since both slopes are equal (2 = 2), points P(1, 2), Q(3, 6), and R(5, 10) are collinear points.

Example 3: Show that the points X(1, 1), Y(2, 3), and Z(3, 5) are collinear using the area method.

Solution: According to the Area Method, if the area of the triangle formed by three points is zero, then the points are collinear.

Step 1: Write down the given points

 X(1, 1), Y(2, 3), Z(3, 5)

Step 2: Formula for area

Area = 1​/2 ∣ x1​(y2​− y3​) + x2​(y3​− y1​) + x3​(y1​− y2​)∣

Area = 1/2 ∣ 1​(3 - 5) + 2(5 -1 ) + 3​(1 - 3)∣

Area = 1/2 ∣ - 2 + 8 - 6∣ = 1/2 ∣0∣  = 0

Since the area is 0, the points X(1, 1), Y(2, 3), and Z(3, 5) are collinear points.

Read More: Mean, Median, Mode

Collinear Points Practice Questions

After learning about what are collinear points,  it’s time to test your understanding with a few simple practice problems. These exercises will help you apply what you’ve learned and recognise collinear points in coordinate geometry.

1. Check if points (2, 3), (4, 7), and (6, 11) are collinear.

2. Verify whether (1, 2), (3, 4), and (5, 8) are collinear.

3. Find if points (0, 0), (4, 8), and (6, 12) are collinear using the slope method.

4. Determine if A(1, -1), B(2, 3), and C(3, 7) are collinear using the area method.

5. Identify three non collinear points that form a right triangle.

Common Collinear Points Mistakes to Avoid 

When solving questions on collinear points, students often make small but important errors that lead to wrong answers. Being careful with your calculations ensures you apply the collinear points formula correctly.

• Forgetting that any two points are always collinear.

• Using the slope formula without checking if the x-values are equal, which can cause division errors.

• Assuming three points are collinear without verifying through calculation.

• Mixing up coordinates while substituting values into the formula.

• Forgetting to take the absolute value in the area (determinant) method.

Also read: Substitution Method

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