Understanding Collinear Points: Methods for Determining Collinearity

collinear points

Collinear points refer to a set of points that all lie on the same straight line

Collinear points refer to a set of points that all lie on the same straight line. In other words, if you can draw a straight line that passes through each point without straying from the line, then the points are said to be collinear.

To determine if a set of points is collinear, you can use the slope formula or determine if the slopes between each pair of points are equal.

Let’s take an example to better understand collinear points:

Imagine you have three points A, B, and C. Point A has coordinates (2, 4), point B has coordinates (4, 8), and point C has coordinates (6, 12).

First, we can calculate the slope between points A and B using the slope formula:
Slope (AB) = (y2 – y1) / (x2 – x1)
= (8 – 4) / (4 – 2)
= 4 / 2
= 2

Next, we calculate the slope between points B and C:
Slope (BC) = (y2 – y1) / (x2 – x1)
= (12 – 8) / (6 – 4)
= 4 / 2
= 2

Since both slopes are equal, we can conclude that points A, B, and C are collinear.

Another method to determine collinearity is by using the concept of area. If the area formed by any set of three points is zero, then the points are collinear. This approach involves calculating the area formed by three points using determinants.

For example, let’s consider three points: A with coordinates (1, 3), B with coordinates (4, 10), and C with coordinates (7, 17).

To calculate the area, we use the determinant of a matrix formed by the coordinates of the points:
Area (ABC) = |1 1 1 |
|1 4 10|
|1 7 17|

Evaluate the determinant:
Area (ABC) = (1 * 4 * 17) + (1 * 10 * 1) + (1 * 7 * 3) – (1 * 4 * 3) – (1 * 1 * 17) – (1 * 7 * 10)
= 68 + 10 + 21 – 12 – 17 – 70
= 0

Since the area formed by points A, B, and C is zero, we can conclude that they are collinear.

In summary, to determine if points are collinear, you can either calculate the slopes between each pair of points and check if they are equal or use the area formed by three points and check if it is zero.

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