Understanding the relationship between perpendicular lines and their slopes is a fundamental concept in geometry and algebra. This article delves into the specifics of how to determine the slope of a line that is perpendicular to another line, using a practical example. We will explore the steps to find the slope of a line perpendicular to the line joining points P (2, -3) and Q (-3, -2).

Introduction to the Problem

Given two points, P (2, -3) and Q (-3, -2), the first task is to find the slope of the line segment PQ.

Step 1: Calculate the Slope of Line Segment PQ

The formula to find the slope of a line joining two points (x1, y1) and (x2, y2) is given by:

$$ m frac{Delta y}{Delta x} frac{y_2 - y_1}{x_2 - x_1} $$

For points P (2, -3) and Q (-3, -2), the slope (m) of the line segment PQ is calculated as follows:

$$ m_{PQ} frac{-2 - (-3)}{-3 - 2} frac{1}{-5} -frac{1}{5} $$

Step 2: Determine the Slope of the Perpendicular Line

A key property of perpendicular lines is that the product of their slopes is -1. Therefore, the slope of the line that is perpendicular to PQ can be found as the negative reciprocal of -1/5.

The negative reciprocal of -1/5 is 5. Thus, the slope of the line perpendicular to PQ is 5.

The formula for the negative reciprocal can also be expressed in a general way. If the slope of a line is m, then the slope of a line perpendicular to it is -1/m. In this case:

$$ m_{perpendicular} -frac{1}{-frac{1}{5}} 5 $$

Real-World Applications and Examples

Understanding the concept of slopes and perpendicular lines is crucial in various fields, including engineering, architecture, and physics. Let's explore a practical application:

Suppose you are designing a path that is perpendicular to an existing path. The slope of the existing path is known to be -1/5. Using the concept discussed, you can easily determine that the slope of the new path must be 5 to ensure it is perpendicular.

Moreover, understanding these concepts helps in determining the equation of a line that passes through a point and is perpendicular to another line. For instance, given the point P (2, -3), the equation of the line perpendicular to PQ with slope 5 can be derived using the point-slope form of the equation of a line:

$$ y - y_1 m(x - x_1) $$

Substituting the point P (2, -3) and the slope 5:

$$ y - (-3) 5(x - 2) $$

Which simplifies to:

$$ y 3 5(x - 2) $$

This is the equation of the line that is perpendicular to PQ and passes through point P.

Conclusion

Perpendicular lines and their slopes play a crucial role in both theoretical and practical applications. By understanding the relationship between the slopes of perpendicular lines and the formula for calculating negative reciprocals, you can effectively solve problems involving perpendicular lines in geometry and algebra.

Whether you are designing paths, analyzing geometric figures, or solving complex algebraic equations, the knowledge of slopes and perpendicular lines is invaluable. By applying the steps discussed in this article, you can confidently find the slope of a line perpendicular to another line, fulfilling the requirements of various mathematical and real-world scenarios.