Collinear Points: Definition, Formula & Real-World Applications

Have you ever noticed how telephone poles, railway tracks, or even planets in an eclipse sometimes appear to be perfectly aligned? This alignment is an example of collinear points in real life. Understanding collinear points is crucial in geometry, as they help in solving problems related to alignment, coordinates, and slopes.

This article will cover the following topics:

• What are collinear points?
• How do you determine if three points are collinear?
• What mathematical formulas help check collinearity?
• Where do collinear points appear in real life?

What Are Collinear Points?

Collinear points are three or more points that lie on the same straight line. In other words, if a single straight line can pass through all given points, they are considered collinear.

Key Properties of Collinear Points:

• Any two points are always collinear because a line can always be drawn through them.
• Three or more points are collinear if they lie on the same straight line.
• The slope between any two pairs of collinear points remains constant.

How to Determine Collinearity?

There are three primary methods to determine if three points are collinear:

1. Using the Slope Formula

Three points (x₁, y₁), (x₂, y₂), and (x₃, y₃) are collinear if the slope between any two pairs is the same:

m₁₂ = m₂₃

Where the slope m between two points is given by:

m = (y₂ − y₁) / (x₂ − x₁)

If the slopes are equal, the points are collinear.

2. Using the Area of a Triangle Method

Three points are collinear if the area of the triangle they form is zero. The area of a triangle given three vertices is:

A = (1/2) |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

If A = 0, the points are collinear.

3. Using Determinants

Using matrix determinants, three points are collinear if:

| x1 y1 1 || x2 y2 1 | = 0| x3 y3 1 | 

Examples of Finding Collinear Points

Example 1:

Determine whether the points (1, 2), (3, 6), and (5, 10) are collinear using the slope method.

Slope between (1,2) and (3,6):

m₁₂ = (6 − 2) / (3 − 1) = 4 / 2 = 2

Slope between (3,6) and (5,10):

m₂₃ = (10 − 6) / (5 − 3) = 4 / 2 = 2

Since m₁₂ = m₂₃, the points are collinear.

Example 2:

Check if the points (2, 4), (6, 8), and (10, 12) are collinear using the area method.

Using the formula:

A = (1/2) |2(8 − 12) + 6(12 − 4) + 10(4 − 8)|

= (1/2) |2(−4) + 6(8) + 10(−4)|

= (1/2) |−8 + 48 − 40|

= (1/2) × 0 = 0

Since the area is 0, the points are collinear.

Practice Questions

1. Determine if the points (0,0), (2,3), and (4,6) are collinear using the slope method.
2. Use the area method to check if (1,1), (3,4), and (5,7) are collinear.
3. Find the value of k if the points (2,3), (4,k), and (6,7) are collinear.

Real-World Applications

• Surveying & Navigation: Used to determine alignment in GPS and mapping.
• Architecture & Engineering: Helps in designing straight beams and alignments in construction.
• Astronomy: Used to track planetary motion and eclipses.
• Computer Graphics: Essential in rendering straight lines in digital imaging.

Understanding collinear points is fundamental in geometry and real-world applications. By using slope, area, or determinant methods, one can easily check collinearity. This concept is widely applied in navigation, physics, engineering, and even digital design.