Determine Collinearity of Points and Construct their Line Equation

In this article, we will explore the concept of collinearity in geometry, specifically focusing on proving that given points are collinear. We will also demonstrate how to construct the equation of the line passing through two of these points and verify if the third point lies on the same line. This is a fundamental topic in geometry and algebra, which enhances our understanding of the relationships between points and lines in a plane.

Introduction to Collinearity

Collinearity refers to the property of points being on the same straight line. In simpler terms, if you can draw a single straight line through a set of points, those points are collinear.

Steps to Prove Points are Collinear

To prove that points A(-2, 5), B(0, 1), and C(2, -3) are collinear, we can use vectors or the slope formula. Let's explore both methods.

Vector Method

The vector method involves finding the vectors between pairs of points and checking if these vectors are scalar multiples of each other. If so, the points are collinear.

Checking Vectors

Let's denote the vectors as follows:

A-B (-2-0, 5-1) (-2, 4)

A-C (-2-2, 5-(-3)) (-4, 8)

B-C (0-2, 1-(-3)) (-2, 4)

We can see that A-B -2(A-C). Therefore, the vectors A-B and B-C are scalar multiples of each other, proving that points A, B, and C are collinear.

Slope Method

To use the slope method, we need to calculate the slopes of the line segments formed by these points and check if they are equal. If the slopes are equal, the points are collinear.

Calculating Slopes

The slope of a line connecting two points (x1, y1) and (x2, y2) is given by:

slope (y2 - y1)/(x2 - x1)

Let's calculate the slopes of the line segments AB, BC, and AC:

slope AB (1 - 5)/(0 - (-2)) -4/2 -2

slope BC (-3 - 1)/(2 - 0) -4/2 -2

slope AC (-3 - 5)/(2 - (-2)) -8/4 -2

Since the slopes of AB, BC, and AC are equal, points A, B, and C are collinear.

Constructing the Equation of the Line Passing Through Two Points

Now that we have proven the points are collinear, we will construct the equation of the line passing through any two of these points, say A(-2, 5) and B(0, 1).

Equation of Line

The general form of the equation of a line is:

y - y1 m(x - x1)

where m is the slope and (x1, y1) is a point on the line. Using point A(-2, 5) and the slope -2, we get:

y - 5 -2(x - (-2))

Simplifying:

y - 5 -2(x 2)

y - 5 -2x - 4

y -2x 1

Multiplying through by -1 to get a more familiar form:

2x y - 1 0

Verification

To verify if point C(2, -3) lies on this line, we substitute its coordinates into the equation:

2(2) (-3) - 1 4 - 3 - 1 0

Since the equation is satisfied, we have verified that point C lies on the line passing through A and B.

Thus, the points A(-2, 5), B(0, 1), and C(2, -3) are indeed collinear, and the equation of the line passing through them is 2x y - 1 0.

Conclusion

This article has provided a comprehensive understanding of how to determine if points are collinear and construct the equation of the line passing through them. By using both vector and slope methods, we ensured a robust proof. The equation of the line we derived can now be used for further geometric or algebraic analysis involving these points.