Any Three Points Are Coplanar

Collinear Points

Collinear points are those points that lie on the same directly line. It is not necessary that they should be co-planar but they must lie on the same directly line. The discussion collinear is derived from the Latin words 'col' and 'linear' where 'col' stands for together and 'linear' ways in the same line. The property of the points beingness collinear is known as collinearity. Let united states of america acquire more about collinear points in this commodity.

1.	What are Collinear Points?
2.	Non-Collinear Points
three.	Collinear Points Formula
iv.	FAQs on Collinear Points

What are Collinear Points?

Collinear points are a set of 3 or more points that be on the same straight line. Collinear points may exist on different planes but not on different lines. The holding of points being collinear is known equally collinearity. And then any three points or more will only exist collinear if they are in the same straight line. Merely one line is possible that can go through three different points which are collinear. Observe the figure given below in which points P, Q, and R are the collinear points.

Collinear points P, Q, and R

Non-Collinear Points

If three or more points do not lie on the same straight line, and then they are said to be non-collinear points. If any point of all the points is not on the same line, then equally a group they are non-collinear points. In the figure given below, points M, Due north, O, P, and Q are non-collinear points since they do not lie on the same directly line.

Non- Collinear points

Collinear Points Formula

The collinear points formula is used to discover out whether three points are collinear or not. There are various methods that tin determine whether three points are collinear or not. The three most mutual formulas that are used to find if points are collinear or not are the Gradient Formula, the Surface area of Triangle Formula, and the Distance Formula. Let united states discuss all these formulas one by one.

Gradient Formula

We apply the slope formula to find the slope of lines formed by the 3 points under consideration. If the iii slopes are equal, and so the three points are collinear.

For case, if nosotros have three points X, Y, and Z, the points will be collinear only if the gradient of line XY = slope of line YZ = slope of line XZ. To calculate the slope of the line joining ii points, we apply the gradient formula.

The slope of the line joining points P(10₁, y₁) and Q(10₂, y₂) is:

\(g = \dfrac{y_{2}-y_{one}}{x_{2}-x_{1}}\)

Surface area of Triangle Formula

In this method, we use the fact that a triangle cannot be formed by 3 collinear points. This means if whatever 3 points are collinear they cannot form a triangle. Therefore, nosotros cheque the points of the triangle by using them in the formula for the area of a triangle. If the expanse is equal to 0, then those points will exist considered to be collinear. In other words, the triangle formed past three collinear points will have no expanse since it will just be a line joining the three points. The formula for the area of a triangle that is used to cheque the collinearity of points is expressed equally:

Area of the triangle with the given points (vertices) A(10_one, y₁), B(x₂, y₂), and C(x₃, y₃) is:

\( \text{A} =\frac{i}{2}\left|\left(x_{1}\left(y_{2}-y_{iii}\correct)+x_{2}\left(y_{3}-y_{one}\correct)+x_{iii}\left(y_{1}-y_{2}\correct)\right)\correct|=0\)

Distance Formula

Using the distance formula, nosotros discover the altitude betwixt the first and the 2nd point, then the altitude betwixt the 2d and the third betoken. After this, we check if the sum of these two distances is equal to the distance between the showtime and the third point. This volition only exist 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 betwixt two points A(x₁, y_ane) and B(10₂, y₂) is:

\(d=\sqrt{\left(x_{2}-x_{1}\right)^{two}+\left(y_{two}-y_{1}\right)^{2}}\)

Then, if we have three collinear points in the guild A, B, and C, then these points volition be collinear if AB + BC = CA.

All the three methods tin can exist understood with the assist of the solved examples given nether the Practice Section.

Tips on Collinear Points:

Three points will exist collinear, merely if they autumn in the same straight line.
This property of points being collinear is known as collinearity.
Collinear points tin exist on dissimilar planes.

☛ Related Topics

X and Y Graph
Coordinate Aeroplane
Equation of a Straight Line
Area of triangle in coordinate geometry
Distance betwixt Two Points
Equation of a Line

Collinear Points Examples

Example 1:

By using the slope formula, observe out whether the points P(1, 2), Q(2, iii), and R(3, 4) are collinear points or non.

Solution:

To bank check the collinearity of points, we will use the slope formula and find the slope of any 2 pairs of lines formed by the points

Allow usa discover the slope of the lines PQ and QR, and check if we get the slopes equal to each other. If they are equal, and so the points will be collinear.

Slope of line PQ is

\(\begin{align*} m_{two} &=\dfrac{y_{2}-y_{ane}}{x_{2}-x_{ane}} \\ m_{ii} &=\dfrac{three-2}{2-1} \\ m_{2} &=\dfrac{1}{1} \\ m_{2} &=one \\ m_{i} &=m_{two} \terminate{align*}\)

Slope of line QR is

\(\begin{marshal*} m_{1} &=\dfrac{y_{3}-y_{2}}{x_{iii}-x_{ii}} \\ m_{1} &=\dfrac{iv-3}{iii-2} \\ m_{1} &=\dfrac{one}{1} \\ m_{1} &=1 \end{align*}\)

Every bit the gradient of both the lines is equal, the points are collinear.

Reply: P(i, ii), Q(ii, 3), and R(three, iv) are collinear points.
Example 2:

Check whether the given points are collinear or not: A (−3, −1), B (−i, 0), and C (1, i).

Solution:

Given:

A(ten₁, y₁) is (-3, −1)
B(x₂, y₂) is (-1, 0)
C(10₃, y₃) is (1, 1)

Let us use the area of triangle formula to check the collinearity of the given points. If the points are collinear, so the area of the triangle volition be 0:
Area of triangle =\(\frac{1}{2} \mid\left(x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{one}\right)+x_{three}\left(y_{1}-y_{2}\right) \mid=0\right.\)

On substituting the values, we become:

\(\dfrac{1}{two} \mid(-3(0-one)+ (-i)(ane-(-1))+ane(-1-0) \mid=0 \\ \dfrac{one}{2}|(3 -2 -1)| =0 \\ 0=0 \)

As the surface area of the triangle is 0, the points are collinear.

Answer: The points A(−3, −1), B (−1, 0), and C (1, 1) are collinear.
Example 3: Use the distance formula to check whether the given points are collinear points or non: P(8, xi), Q(2, three), R(-1, -1).

Solution: If the distance of PQ + altitude of QR = distance of PR, then the 3 points P, Q, and R will be collinear. And then, allow us find the distance betwixt the points P and Q using the distance formula where x₁ = eight, y₁ = 11, 10₂ = ii, y₂ = 3

PQ = √[(x₂ - 10₁)^two + (y₂- y_i)²]

PQ = √[(2 - eight)² + (3- 11)ⁱⁱ]

PQ = √[(- 6)ⁱⁱ + (-8)^two]

PQ = √(36 +64) = √100 = 10

At present, allow u.s. find the distance between Q and R using the distance formula where, x₁ = 2, y₁ = 3, 10₂ = -1, y_two = -1

QR = √[(x₂ - ten₁)² + (y_two- y_ane)^two]

QR = √[(-1 - ii)^two + (-1- 3)²]

QR = √[(-iii)² +(-4)²)] = √25 = v

PQ + QR = 10 + v = 15 units

At present, let us discover the distance between P and R using the altitude formula where, ten₁ = viii, y₁ = 11, x_two = -one, y_ii = -1

PR = √[(-ane - 8)² + (-one- eleven)²]

PR = √[(-nine)² + (-12)²]

PR = √(81 +144) = √225 = 15 units

Therefore, information technology can be seen that PQ + QR = 10 + 5 = 15 units, and the distance betwixt P and R = 15 units. This means that the given points are collinear.

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Practise Questions on Collinear Points

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FAQs on Collinear Points

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 dissimilar planes but not on different lines.

How to Find Collinear Points?

There are various methods that are used to observe out whether three points are collinear or non. The 3 almost common methods used to find out the collinearity of points is by using the altitude formula, the slope formula, and the area of triangle formula. Using these formulas, we find out whether the points are collinear or non. A detailed explanation is given under the heading 'Collinear Point Formula' on this page.

Altitude Formula: Using the altitude formula, we detect the distance between the first and the 2nd point, and so the distance betwixt the second and the third point. And so, nosotros check if the sum of these two distances is equal to the distance between the kickoff and the third point. This volition only exist possible if the three points are collinear points.
Gradient Formula: Nosotros utilize the gradient formula to observe the slope of the lines formed by the 3 points under consideration. If the three slopes are equal, then the three points are collinear.
Expanse of Triangle Formula: Using the expanse of the triangle formula, we utilize the fact that a triangle formed by three collinear points will have no area since information technology volition just be a line joining the three points. Therefore, we check the points of the triangle by using them in the formula for the area of a triangle. If the area is equal to 0, then those points will be considered to exist collinear.

How to Name Collinear Points?

Collinear points are named in the usual fashion that is used to label whatever point using capital letters, notwithstanding, they are mentioned together equally a group. This means if points A, B, and C form a direct line and so we say that points A,B, and C are collinear.

How to Show that 3 Points are Collinear?

There are many ways in which any 3 points can be proved to be collinear. I of the methods is by using the formula for the expanse of a triangle. We substitute the coordinates of all the 3 points in this formula. If the area comes to 0, this proves that the three points are collinear. The formula that is used is,

\( \text{A} =\frac{1}{ii}\left|\left(x_{1}\left(y_{2}-y_{3}\right)+x_{two}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right)\correct|=0\)

where the given points A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) are the vertices of the triangle.

What are the Collinear Points in a Triangle?

According to the Euler's theory, in a triangle, there exists a direct line called the Euler's line, which passes through the orthocenter, the circumcenter, and the centroid of the triangle. Hence, these given points of concurrencies of the triangle are the collinear points in a triangle.

What Does Collinear Mean?

The word collinear is derived from the Latin words 'col' and 'linear' where col stands for together and linear means in the same line. Collinear points are the grouping of three or more than than iii points that lie on the same straight line. It is not necessary that they are coplanar merely they must prevarication on the same straight line.

Are 2 Points E'er Collinear?

Aye, two points are ever collinear since we can draw a straight line between any two points. There exist no 2 such points through which a straight line cannot laissez passer. Therefore, whatever 2 points are always collinear points.

What are Non-Collinear Points?

If 3 or more than points do not lie on the same directly line, then they are said to be non-collinear points. If any point of the given points is not on the aforementioned line, and then as a group they are not-collinear points. For not-collinear points, the area of the triangle joined by the three points will always be greater than 0.

What is the Difference Between Collinear and Non-Collinear Points?

Collinear points are two or more points that prevarication on a straight line whereas not-collinear points are points that do non lie on 1 direct line.