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.

Collinear Points Definition

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

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.

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.

Formula for Collinear points

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

  • Slope Formula
  • Area of triangle

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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