Determining Collinear Points: Slope Formula and Equation Method

collinear points

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

Collinear points refer to a set of points that lie on the same straight line. In other words, if you can draw a line that passes through all the given points, then those points are said to be collinear.

To determine if a set of points are collinear, you can use the slope formula or the method of finding the equation of a line.

Using the slope formula, you can calculate the slope between two pairs of points. If the slopes of all pairs of points are equal, then the points are collinear. The slope formula is given by:

m = (y2 – y1) / (x2 – x1)

Let’s consider an example to understand this better. Suppose we have three points A(2, 4), B(6, 10), and C(8, 16). We can find the slopes between these pairs of points:

Slope of AB:
mAB = (10 – 4) / (6 – 2) = 6 / 4 = 3/2

Slope of AC:
mAC = (16 – 4) / (8 – 2) = 12 / 6 = 2

Slope of BC:
mBC = (16 – 10) / (8 – 6) = 6 / 2 = 3

As we can see, the slopes of AB, AC, and BC are not equal. Therefore, the points A, B, and C are not collinear.

Alternatively, you can also use the method of finding the equation of a line. If the equations of two lines passing through pairs of points are the same, then the points are collinear.

For example, let’s consider the points A(1, 2), B(4, 6), and C(7, 10). We can find the equations of the lines passing through AB and AC:

Equation of line AB:
y – y1 = (y2 – y1) / (x2 – x1)(x – x1)
=> y – 2 = (6 – 2) / (4 – 1)(x – 1)
=> y – 2 = 4/3(x – 1)
=> y – 2 = 4/3x – 4/3
=> y = 4/3x + 2/3

Equation of line AC:
y – y1 = (y2 – y1) / (x2 – x1)(x – x1)
=> y – 2 = (10 – 2) / (7 – 1)(x – 1)
=> y – 2 = 8/6(x – 1)
=> y – 2 = 4/3x – 4/3
=> y = 4/3x + 2/3

As we can see, the equations of the lines AB and AC are the same. Therefore, the points A, B, and C are collinear.

In conclusion, collinear points are points that lie on the same straight line. You can determine if points are collinear by checking if the slopes between pairs of points are equal or by verifying if the equations of lines passing through the points are the same.

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