How To Find If Three Points Are Collinear

In geometry, a betoken determines the location on a aeroplane. Nosotros can mark any number of points on a airplane. Suppose if you lot marker three points on a paper, nosotros would be required to characterization them by a single majuscule like A, B, C or P, Q, R since we need to represent the points using upper-case letter letters. Also, nosotros tin can draw several shapes that pass through these three points such equally a line, ray, line segment when they are collinear points otherwise, we tin can depict a triangle or circle, etc. Here, yous can learn how to find collinearity of iii points in dissimilar ways using gradient, altitude, area, and and then on.

Condition of Collinearity of 3 Points

In general, lines can be parallel, perpendicular, intersection, etc. In all these cases, the slopes of lines are related to each other in some way. As we know, the slopes of ii parallel lines are equal. If two lines take the same slope laissez passer through a common point, and so two lines will coincide. In other words, if A, B, and C are iii points in the XY-plane, they volition lie on a line, i.e., three points are collinear if and only if the slope of AB is equal to the slope of BC.

This scenario can be observed in the beneath effigy.

Collinearity of three points

From the to a higher place, we tin can derive the condition for collinearity of three points A, B and C using slope formula.

Allow (x₁, y₁), (x₂, y₂) and (x₃, y₃) be the coordinates of three points, say A, B and C, respectively.

Slope of AB = Gradient of BC {that means A, B and C are collinear}

(y₂ – y_one)/ (10_ii – 10_one) = (y_iii – y₂)/ (x₃ – x_two)

(10₃ – x_ii)(y_ii – y_i) = (x_two – x_one)(y_iii – y₂)

x₃(y₂ – y₁) – ten_two(y_two – y₁) = x₂(y_three – y_two) – x_ane(y_three – y_ii)

Rearranging the terms,

x₂(y₃ – y_ii) + x₁(y₂ – y_three) + ten₂(y₂ – y₁) – 10₃(y₂ – y₁) = 0

x_ane(y₂ – y₃) + x₂(y_iii – y_two + y₂ – y₁) + ten_three(y_i – y₂) = 0

x₁(y_ii – y₃) + x₂(y_three – y₁) + x_iii(y₁ – y_ii) = 0

This is known as the collinearity of three points formula.

How to Prove Collinearity of 3 Points

The following conditions are used to prove the collinearity of given points.

Suppose the points A(x_one, y₁), B(x₂, y₂) and C(x_three, y₃) are collinear, then the Conditions for Collinearity of Three Points are:

(i) Gradient of AB = Gradient of BC

(ii) AB + BC = Air-conditioning (or) AB + AC = BC (or) AC + BC = AB

This tin exist proved using the distance formula in coordinate geometry.

(three) Area of triangle ABC = 0, i.e. (½)|10₁(y₂ – y₃) + 10₂(y_three – y_one) + x₃(y_one – y₂)| = 0

(iv) If the third point satisfies the equation through any 2 of the given three points, and then the three points A, B, and C will be collinear.

Solved Examples

Case 1:

Find the value of p for which the points (p, -1), (2, 1) and (4, five) are collinear.

Solution:

Permit the given points be:

A(p, -1) = (x_ane, y₁)

B(two, 1) = (10_ii, y₂)

C(iv, 5) = (10₃, y_iii)

Given that A, B, and C are collinear.

Gradient of AB = Slope of BC

(y_ii – y_one)/(x_ii – x₁) = (y₃ – y₂)/(x₃ – x_ii)

Substituting the values of coordinates of given points,

(1 + 1)/(ii – p) = (five – 1)/(4 – 2)

2/(2 – p) = 4/2

2/(two – p) = 2

⇒ 2 – p = i

⇒ p = 2 – 1

⇒ p = 1

Hence, the value of p is 1.

Instance two:

Using the equation method, check the collinearity of the points A(7, -ii), B(ii, iii) and C(-ane, 6).

Solution:

Nosotros know that the equation of a line passing through the points (10_one, y₁) and (x_two, y_ii) is:

y – y₁ = [(y₂ – y₁) /(x₂ – x₁)] (10 – x_one)

Let A(vii, -ii) = (x₁, y₁) and B(2, 3) = (x₂, y₂).

So, the equation of a line passing through the points A(seven, -2) and B(ii, 3) is given past:

y + ii = [(3 + 2)/(2 – 7)] (x – 7)

y + two = (5/-5) (10 – 7)

y + 2 = -ten + 7

x + y + 2 – vii = 0

x + y – 5 = 0

Now, substituting the indicate C(-1, 6) in the above equation,

-ane + 6 – 5 = 0

0 = 0

Thus, the tertiary point satisfies the equation of the line passing through the two of given iii points.

Therefore, the given points A, B and C are collinear.

Frequently Asked Questions – FAQs

How do you observe Collinearity with 3 points?

We tin find the collinearity with three points in different methods such as:
(i) Using gradient formula
(ii) Using distance formula
(iii) Using area of triangle formula
(iv) Using equation method

What are 3 collinear points?

The points A, B and C are collinear points if they lie on the same line in a plane. In that case, the gradient of AB is equal to the slope of BC.

What is the formula of collinear points?

If the points (10₁, y₁), (x₂, y₂) and (x₃, y₃) are collinear, and so the below formula can be given:
x_one(y₂ – y₃) + 10₂(y₃ – y₁) + x_iii(y₁ – y₂) = 0

Are three points ever collinear?

No, the given three points cannot always be collinear. That means, they can exist collinear or noncollinear depending on the position of these points in a plane.

How do y'all testify collinearity of three points using the altitude formula?

Suppose three points A, B, and C are collinear, then using distance formula we can show the collinearity of these points when they satisfy either of the following atmospheric condition:
(i) AB + BC = Air-conditioning
(two) AB + Air-conditioning = BC
(iii) Air conditioning + BC = AB