Proving Collinearity of Points with Slope and Area Methods: A Comprehensive Guide

Collinearity is a fundamental concept in geometry dealing with the mutual alignment of points on a straight line. In this article, we explore two methods to prove that the points A2,5, B0,3, and C6,-6 are collinear: the slope method and the area method of a triangle formed by these points.

Slope Method

To determine if points are collinear, we can use the slope between each pair of points. If the slopes are equal, the points lie on the same straight line.

Step 1: Calculate the Slope between Points A and B

The formula for the slope m between two points x1, y1 and x2, y2 is: m (y2 - y1) / (x2 - x1)

Slope between A2,5 and B0,3 (mAB): mAB (3 - 5) / (0 - 2) (-2) / (-2) 1 Slope between B0,3 and C6,-6 (mBC): mBC (-6 - 3) / (6 - 0) (-9) / (6) -3/2 Slope between A2,5 and C6,-6 (mAC): mAC (-6 - 5) / (6 - 2) (-11) / (4) -11/4

Since mAB 1, mBC -3/2, and mAC -11/4 are not equal, the points A, B, and C are not collinear.

Alternative Method: Area of Triangle

An alternative method to check collinearity is by determining if the area of the triangle formed by the three points is zero. The formula for the area A is:

A 1/2(x1y2 - y3, x2y3 - y1, x3y1 - y2)

Substitute the coordinates of points A2,5, B0,3, and C6,-6 into this formula:

A 1/2((2*3) - (-6), (0*(-6)) - 5, (6*5) - 3) 1/2(6 6, 0 - 5, 30 - 3) 1/2(12, -5, 27) 1/2 * 30 15

Since the area is not zero, the points A, B, and C are not collinear.

Deriving the Equation of Line AB and Testing Point C

We can derive the equation of line AB to further confirm if point C lies on this line.

Step 1: Find the Equation of Line AB

The slope m between A2,5 and B0,3 is 1, so the equation of line AB is:

y x c

Step 2: Find the y-intercept c

Substitute the coordinates of point A into the equation to find c:

5 2 c

c 3

The equation of line AB is y x 3.

Step 3: Test Point C on the Line

Substitute the coordinates of point C into the equation of line AB:

-6 ≠ 6 3

Since -6 does not equal 9, point C does not lie on line AB, confirming that points A, B, and C are not collinear.

Conclusion: The points A2,5, B0,3, and C6,-6 are not collinear, as demonstrated using both the slope method and the area method of the triangle.