The Points R23 and S-56 on a Straight Line: Understanding the Gradient

Calculating the gradient of a line is an essential concept in coordinate geometry. The gradient (or slope) of a line is defined as the rate of change of the y-coordinates with respect to the x-coordinates. In other words, it quantifies the steepness of the line.

Formula for Gradient

The formula for the gradient, often denoted as ( r ) or ( m ), between two points ( A(x_1, y_1) ) and ( B(x_2, y_2) ) is given by:

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

This formula allows us to determine the change in y-coordinates (rise) over the change in x-coordinates (run), providing a clear measure of the line's steepness.

Example: Points R23 and S-56

Let's consider the points R(2, 3) and S(-5, -6) on a straight line. Using the formula for gradient:

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

By substituting the given coordinates:

r (frac{-6 - 3}{-5 - 2})

This simplifies to:

r (frac{-9}{-7})

Which further simplifies to:

r (frac{3}{7})

Thus, the gradient of the line passing through the points R23 and S-56 is (frac{3}{7}).

Understanding the Gradient Calculation

To further understand this process, let's break down the calculation:

Step 1: Identify the coordinates of the points.

Point R has coordinates (2, 3) and point S has coordinates (-5, -6).

Step 2: Apply the gradient formula.

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

In this case, ( y_1 3 ), ( y_2 -6 ), ( x_1 2 ), and ( x_2 -5 ).

m (frac{-6 - 3}{-5 - 2})

m (frac{-9}{-7})

m (frac{3}{7})

Therefore, the gradient of the line RS is (frac{3}{7}).

Conclusion

Understanding how to calculate the gradient of a line from given points is crucial in coordinate geometry. By utilizing the gradient formula, we can accurately determine the steepness of a line, providing valuable information in various applications, including computer graphics, physics, and engineering.

Whether you're dealing with coordinates in a two-dimensional plane or a real-world scenario where lines are prevalent, the gradient calculation remains a fundamental tool in any mathematician's or scientist's toolkit.