Collinear Points

Collinear points are the points that lie on the same line. If two or more than two points lie on a line close to or far from each other, then they are said to be collinear, in Euclidean geometry.

The term collinear is the combined word of two Latin names ‘col’ + ‘linear’. ‘Col’ means together and ‘Linear; means line. Therefore, collinear points mean points together in a single line. You may see many real-life examples of collinearity such as a group of students standing in a straight line, a bunch of apples kept in a row, next to each other, etc.

Definition

In geometry, two or more points are said to be collinear, if they lie on the same line.

Collinear Meaning

In more astonishing observation, the term collinear has been used for straightened things, which means, something being “in a row” or “in a line”. 

If we say, that three objects are placed in a line, that means the objects are collinear.

In more astonishing observation, the term collinear has been used for straightened things, that means, something being “in a row” or “in a line”.

Non-Collinear Points: The points which do not lie on the same line are called non-collinear points.

What are Collinear points in Maths?

From the above definition, it is clear that the points which lie on the same line are collinear points. To understand this concept clearly, consider the below figure and try to categorize the collinear and non-collinear points.

Collinear Points

In the above figure, the set of collinear points are {A, D}, {A, C, F}, {A, P, R}, {Q, E, R} and {F, B, R}. The remaining points are said to be non-collinear, i.e. {P, B}, {C, E} and so on.

Non-Collinear Points: The points which do not lie on the same line are called non-collinear points.

Formula

There are three methods to find the collinear points. They are:

  • Distance Formula
  • Slope Formula
  • Area of triangle

Using Distance Formula

If P, Q and R are three collinear points, then,

Distance from P to Q + Distance from Q to R = Distance from P to R

PQ + QR = PR

Now, by the distance formula we know, the distance between two points (x1, y1) and (x2, y2) is given by;

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

Hence, we can easily find the distance between the points P, Q and R, with the help of this formula. 

Example

Find if the points P(−3,−1), Q(−1,0), and R(1,1) are collinear.

Solution: The points P, Q and R are collinear, if;

(Distance between P and Q) + (Distance between Q and R) = Distance between P and R

By Distance formula, we can find the distance between two points.

So,

\(\begin{aligned} \text { Distance between } \mathrm{PQ} &=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} \\ &=\sqrt{(-1+3)^{2}+(0+1)^{2}} \\ &=\sqrt{5} \end{aligned}\)

\(\begin{aligned} \text { Distance between Q and R } &=\sqrt{\left(x_{3}-x_{2}\right)^{2}+\left(y_{3}-y_{2}\right)^{2}} \\ &=\sqrt{(1+1)^{2}+(1-0)^{2}} \\ &=\sqrt{5} \end{aligned}\)

\(\begin{aligned} \text { Distance between P and R } \mathrm{AC} &=\sqrt{\left(x_{3}-x_{1}\right)^{2}+\left(y_{3}-y_{1}\right)^{2}} \\ &=\sqrt{(1+3)^{2}+(1+1)^{2}} \\ &=\sqrt{20}\\ &=2 \sqrt{5} \end{aligned}\)

Hence we can conclude that,

√5 + √5 = 2√5

PQ + QR = PR

Therefore, P, Q and R are collinear.

Using Slope Formula

Three or more points are said to be collinear if the slope of any two pairs of points is the same. The slope of the line basically measures the steepness of the line.

Suppose, X, Y and Z are the three points, with which we can form three sets of pairs, such that, XY, YZ and XZ are three pairs of points. Then, as per the slope formula,

If Slope of XY = Slope of YZ = Slope of XZ, then the points X, Y and Z are collinear.

Note: Slope of the line segment joining two points say (x1, y1) and (x2, y2) is given by the formula:

m = (y2 – y1)/ (x2 – x1)

Example

Show that the three points P(2, 4), Q(4, 6) and R(6, 8) are collinear.

Solution: If the three points P(2, 4), Q(4, 6) and R(6, 8) are collinear, then slopes of any two pairs of points, PQ, QR & PR will be equal.

Now, using slope formula we can find the slopes of the respective pairs of points, such that;

Slope of PQ = (6 – 4)/ (4 – 2) = 2/2 = 1

Slope of QR = (8 – 6)/ (6 – 4) = 2/2 = 1

Slope of PR = (8 – 4) /(6 – 2) = 4/4 = 1

As we can see, the slopes of all the pairs of points are equal.

Therefore, the three points P, Q and R are collinear.

Using the Area of Triangle Formula

If the area of triangle formed by three points is zero, then they are said to be collinear. It means that if three points are collinear, then they cannot form a triangle.

Suppose, the three points P(x1, y1), Q(x2, y2) and R(x3, y3) are collinear, then by remembering the formula of area of triangle formed by three points we get;

\(\frac{1}{2}\begin{vmatrix} x_{1}-x_{2} & x_{2}-x_{3} \\ y_{1}-y_{2} & y_{2}-y_{3} \end{vmatrix}=0\)

Or

(1/2) | [x1(y2 – y3) + x2(y3 – y1) + x3[y1 – y2]| = 0

Example

Find if P(2, 3), Q(4, 0) and R(6, -3) are collinear points.

Solution: As per the area of triangle formula for three coordinates in a plane,

Area = \(\frac{1}{2}\begin{vmatrix} 2-4 & 4-6 \\ 3-0 & 0+3 \end{vmatrix}\\ = \frac{1}{2}\begin{vmatrix} -2 & -2\\ 3 & 3 \end{vmatrix}\)

Area = ½ (6 – 6)

Area = 0

Hence, the points P, Q and R are collinear.

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Frequently Asked Questions on Collinear Points

What are collinear points?

When two or more points lie on the same line, they are called collinear points.

Can we draw a straight line through collinear points?

Yes, we can draw a straight line through collinear points.

What are non-collinear points?

When the points in a place, does not lie on the same line, then such points are called non-collinear points.

How can we determine if the points are collinear?

There are three methods to find if the points are collinear, they are:
Distance formula
Slope formula
Area of triangle

If three points are collinear then slope of points formed by these points are equal. True or False?

True, if three points are collinear then slope of points formed by these points are equal.

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