Understanding Collinear Points: Using Slope and Equations to Determine Collinearity

collinear points

Collinear points are points that lie on the same straight line

Collinear points are points that lie on the same straight line. In simpler terms, three or more points are said to be collinear if a straight line can be drawn through them without any of the points deviating from the line.

To determine if three points are collinear, you can use the slope formula or the equation of a line. Let’s consider three points A, B, and C.

Using the slope formula, you can calculate the slope between the two pairs of points (A and B, and B and C). If the slope between AB is equal to the slope between BC, then the points A, B, and C are collinear.

For example, if the coordinates of points A, B, and C are (1, 2), (3, 4), and (5, 6) respectively, we can calculate the slopes.

Slope of AB (m1) = (y2 – y1) / (x2 – x1)
= (4 – 2) / (3 – 1)
= 2 / 2
= 1

Slope of BC (m2) = (y2 – y1) / (x2 – x1)
= (6 – 4) / (5 – 3)
= 2 / 2
= 1

Since m1 = m2, the points A, B, and C are collinear.

Another way to determine collinearity is using the equation of a line. You can find the equation of the line passing through two of the points (for example, A and B), and then substitute the coordinates of the third point (C) into the equation. If the equation holds true, then the points are collinear.

For instance, if we use points A(1, 2) and B(3, 4) to find the equation of the line and substitute the coordinates of C(5, 6), the equation should still be valid.

The equation of a line in slope-intercept form is y = mx + c, where m is the slope and c is the y-intercept. To find the equation, we need to calculate the slope (m) and y-intercept (c).

m = (y2 – y1) / (x2 – x1)
= (4 – 2) / (3 – 1)
= 2 / 2
= 1

Now, substituting the coordinates of point A (1, 2) into the equation y = mx + c:

2 = 1(1) + c
2 = 1 + c
c = 2 – 1
c = 1

Thus, the equation of the line passing through points A and B is y = x + 1.

Now, if we substitute the coordinates of point C (5, 6) into the equation:

6 = 5 + 1
6 = 6

Since the equation holds true, points A, B, and C are collinear.

In summary, to determine if three points (A, B, and C) are collinear, you can calculate the slopes between AB and BC. If the slopes are equal, the points are collinear. Alternatively, you can find the equation of the line passing through two of the points and check if the third point satisfies the equation.

More Answers: