Understanding Collinearity in Mathematics: Slopes and Areas of Points on a Straight Line

collinear

In mathematics, the term “collinear” refers to a set of points that lie on the same straight line

In mathematics, the term “collinear” refers to a set of points that lie on the same straight line. When three or more points are collinear, it means that they can all be connected by a single straight line without any curves or angles.

To determine if points are collinear, you can use a few different methods. One common method is the slope formula. Suppose you have three points A, B, and C. To check if they are collinear, calculate the slopes of the line segments AB and BC. If the slopes of both line segments are equal, then the points are collinear.

The slope of a line segment AB is calculated using the formula:
m = (y2 – y1) / (x2 – x1)

For example, let’s say we have three points A(1, 2), B(-2, 4), and C(4, 6). To determine if they are collinear, we can calculate the slopes of AB and BC:

Slope of AB = (4 – 2) / (-2 – 1) = 2 / -3 = -2/3
Slope of BC = (6 – 4) / (4 – (-2)) = 2 / 6 = 1/3

Since the slopes of AB and BC are not equal (-2/3 ≠ 1/3), the points A, B, and C are not collinear.

Another method to check for collinearity is by using the concept of area. If the area of the triangle formed by three points is zero, then the points are collinear.

To calculate the area of a triangle, you can use the formula:

Area = 1/2 * |(x1*(y2 – y3) + x2*(y3 – y1) + x3*(y1 – y2))|

For example, let’s take the same points A(1, 2), B(-2, 4), and C(4, 6):

Area = 1/2 * |(1*(4 – 6) + (-2)*(6 – 2) + 4*(2 – 4))|
Area = 1/2 * |-2 – 8 – 4|
Area = 1/2 * |-14|
Area = 1/2 * 14
Area = 7

Since the area of the triangle is not zero (Area ≠ 0), the points A, B, and C are not collinear.

Overall, to determine if points are collinear, you can either check if the slopes of the line segments are equal or calculate the area of the triangle formed by the points. If the slopes are equal or the area is zero, then the points are collinear.

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