What are Collinear Points in Geometry – Definition, Properties & Examples

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A point is one of the basic and fundamental shapes in mathematics. A point can be of different types depending on its position relative to other points or figures. One such category of points is collinear points.

Let’s understand what are collinear points in geometry and their properties with examples.

What are Collinear Points in Geometry?

The term collinear is derived from two Latin words ‘col’ and ‘linear’. ‘Col’ means together and ‘Linear’ means line. Therefore, collinear points are the points that lie on a single line.

You can find many real-life examples of collinearity such as a group of students standing in a straight line, books kept in a row in a library, items arranged in a row in a grocery store, etc.

In geometry, two or more points are said to be collinear, if they lie on the same line. Hence the collinear points are the set of points that lie on a single straight line.

Examples

In the figure below, the three points A, B, and C are called collinear points as they all lie on a single line. In other words, we can say that we can draw a line passing through all three points.

Note:

• Two points are always collinear points.
• A triangle cannot be formed using three collinear points.

Let’s consider one more figure.

In the above figure, the points P, Q, and R are non-collinear points, as they all three are not lying on a single line. In other words, we can say that we cannot draw a line passing through all three points.

Note: Three non-colinear points always form a triangle.

How to Check Three Points Are Collinear?

As discussed above, three points are called collinear points, if they all lie on the same line. In order to check, whether the three given points are collinear, any of the following three methods can be used.

• Distance Formula Method
• Slope Formula Method
• Area of Triangle Method

Distance Formula Method

In this method, we find the distance between the first and the second point, and then the distance between the second and the third point. After this, we check if the sum of these two distances is equal to the distance between the first and the third point. This will only be possible if the three points are collinear points.

To calculate the distance between two points whose coordinates are known to us, we use the distance formula.

The distance between two points $\text{A} \left(x_1, y_1 \right)$ and $\text{B} \left(x_2, y_2 \right)$ is $\sqrt{\left(x_2 – x_1 \right)^2 + \left(y_2 – y_1 \right)^2}$

So, if we have three collinear points in the order A, B, and C, then these points will be collinear if AB + BC = CA.

Let’s consider a few examples to understand the method.

Examples

Ex 1: Check whether the points $\text{P}(−3,−1)$, $\text{Q}(−1,0)$, and $\text{R}(1,1)$ are collinear.

Distance between the points P and Q = |PQ| = $\sqrt{\left( -1 – (-3)\right)^2 + \left(0 – (-1) \right)^2}$

$= \sqrt{\left( -1 + 3\right)^2 + \left(0 + 1 \right)^2}$

$= \sqrt{2^2 + 1^2}$

$= \sqrt{4 + 1}$

$= \sqrt{5} \text{ units}$

Distance between the points Q and R = |QR| = $\sqrt{\left(1 – (-1)\right)^2 + \left(1 – 0 \right)^2}$

$= \sqrt{\left(1 + 1\right)^2 + \left(1 – 0 \right)^2}$

$= \sqrt{2^2 + 1^2}$

$= \sqrt{4 + 1}$

$= \sqrt{5} \text{ units}$

Distance between the points P and R = |PR| = $\sqrt{\left(1 – (-3)\right)^2 + \left(1 – \left(-1 \right) \right)^2}$

$= \sqrt{\left(1 + 3\right)^2 + \left(1 + 1 \right)^2}$

$= \sqrt{\left(4\right)^2 + \left(2 \right)^2}$

$= \sqrt{16 + 4}$

$= \sqrt{20}$

$= 2\sqrt{5} \text{ units}$

Note that $\sqrt{5} + \sqrt{5} = 2\sqrt{5}$, therefore, PQ + QR = PR, which means that the three points P, Q, and R are collinear points.

Ex 2: Check whether the points $\text{A}(1,0)$, $\text{B}(0,0)$, and $\text{C}(0,1)$ are collinear.

Distance between the points A and B = |AB| = $\sqrt{\left( 0 – 1\right)^2 + \left(0 – 0 \right)^2}$

$= \sqrt{\left(- 1\right)^2 + 0^2}$

$= \sqrt{1 + 0}$

$= \sqrt{1}$

$= 1 \text{ units}$

Distance between the points B and C = |BC| = $\sqrt{\left(0 – 0\right)^2 + \left(1 – 0 \right)^2}$

$= \sqrt{0^2 + 1^2}$

$= \sqrt{0 + 1}$

$= \sqrt{1}$

$= 1 \text{ units}$

Distance between the points A and C = |AC| = $\sqrt{\left(0 – 1\right)^2 + \left(1 – 0 \right)^2}$

$=\sqrt{\left(- 1\right)^2 + 1^2}$

$=\sqrt{1 + 1}$

$=\sqrt{2} \text{ units}$

Since the length AC is greater than the other two, therefore, we will add the other two distances, i.e., AB and BC.

AB + BC = $1 + 1 = 2 \text{ units}$

Practice Problems

1. What are collinear points?
2. What are non-collinear points?
3. What are the three methods to check whether the three points are collinear?
4. Check whether the points P(1, 2), Q(2, 3), and R(3, 4) are collinear points or not using
• distance formula method
• slope formula method
• area of the triangle method
5. Check whether the points P(-3, -1), Q(-1, 2), and R(1, 1) are collinear points or not using
• distance formula method
• slope formula method
• area of the triangle method

FAQs

What are collinear points in geometry?

Collinear points are a set of three or more points that exist on the same straight line. Collinear points may exist on different planes but not on different lines.

How to find collinear points?

There are various methods that are used to find out whether the three points are collinear or not. The three most common methods used to find out the collinearity of points is by using the distance formula, the slope formula, and the area of triangle formula. Using these formulas, we find out whether the points are collinear or not.

Conclusion

Collinear points are a set of three or more points that exist on the same straight line. The points that are not collinear are called non-collinear points. The three methods to check the collinearity of three points are using the distance formula, the slope formula, and the area of triangle formula.