Determining if Points Lie on a Straight Line Using Slopes

Understanding if points (0, y1), (1, y2), and (2, y3) lie on a straight line is a fundamental concept in analytical geometry. This article explores the conditions and methods required to determine if these points form a straight line using the concept of slope. This knowledge is crucial for various applications, from basic geometry to more advanced mathematical analysis.

Understanding Collinear Points

In geometry, collinear points are points that lie on the same straight line. To verify if the points (0, y1), (1, y2), and (2, y3) are collinear, we need to ensure that the slope between any two of these points is the same. The slope between two points (x1, y1) and (x2, y2) is given by the formula:

Slope ( frac{y_2 - y_1}{x_2 - x_1} )

Calculating Slopes

To ensure that (0, y1), (1, y2), and (2, y3) are collinear, we need to calculate the slopes between all pairs of these points. Specifically, we need to verify that the slope between (0, y1) and (1, y2) is equal to the slope between (1, y2) and (2, y3).

The slope between (0, y1) and (1, y2) is:

Slope01 ( frac{y_2 - y_1}{1 - 0} y_2 - y_1 )

The slope between (1, y2) and (2, y3) is:

Slope12 ( frac{y_3 - y_2}{2 - 1} y_3 - y_2 )

The slope between (0, y1) and (2, y3) is:

Slope02 ( frac{y_3 - y_1}{2 - 0} frac{y_3 - y_1}{2} )

For the points to be collinear, the above slopes must be equal. This can be mathematically written as:

Slope01 Slope12

This simplifies to:

y2 - y1 y3 - y2

Further rearranging, we get:

2y2 y1 y3

Or:

y2 ( frac{y_1 y_3}{2} )

Arithmetic Sequence and Collinearity

This condition ensures that the y-coordinates of the points form an arithmetic sequence, meaning the y-coordinates are equidistant along the line. Hence, if the points (0, y1), (1, y2), and (2, y3) lie on a straight line, they must satisfy the arithmetic sequence condition:

y2 ( frac{y_1 y_3}{2} )

Geometric Representation and Slopes

The continuous and consistent rate of change between any two points on a straight line ensures that the slope remains constant. A mathematical representation of this is:

y2 - y1 ( frac{y_3 - y_2}{2 - 1} )

This equality demonstrates that the rate of change between the y-coordinates is linear, confirming the collinearity of the points.

Conclusion

To summarize, the points (0, y1), (1, y2), and (2, y3) lie on a straight line if and only if the following condition is satisfied:

y2 ( frac{y_1 y_3}{2} )

This concept is crucial for understanding and visualizing collinear points in geometry and has applications in various fields such as physics, engineering, and data analysis. By ensuring the slopes are equal, we can verify the collinearity of points and leverage this knowledge for more advanced mathematical and practical applications.