Introduction

In this detailed guide, we will walk through the process of finding the equation of a line that passes through a given point and is perpendicular to a line segment joining two other points. This is a fundamental concept in both Geometry and Algebra. By understanding the principles involved, you will be able to solve similar problems with ease, making your work more efficient and accurate.

Understanding Perpendicular Lines

If you are familiar with the concept of slope in a linear equation, you know that it represents the change in y over the change in x. When two lines are perpendicular, their slopes are negative reciprocals of each other. This means if one line has a slope (m_1), the line perpendicular to it will have a slope (m_2 -frac{1}{m_1}).

Step-by-Step Guide

Step 1: Find the Slope of the Given Line Segment

Let's start by finding the slope of the line segment joining points A (3, -1) and B (-2, 2).

The formula for the slope (m) of a line segment joining two points ((x_1, y_1)) and ((x_2, y_2)) is:

[m frac{y_2 - y_1}{x_2 - x_1}]

Substituting the coordinates of points A and B:

[m_{AB} frac{2 - (-1)}{-2 - 3} frac{2 1}{-5} -frac{3}{5}]

Step 2: Determine the Slope of the Perpendicular Line

The slope of a line perpendicular to the line segment joining A and B is the negative reciprocal of the slope we just calculated. Therefore:

[m_{perpendicular} -left(-frac{5}{3}right) frac{5}{3}]

Step 3: Use the Point-Slope Form to Find the Equation

The point-slope form of a line is given by:

[y - y_1 m(x - x_1)]

Using point P (-1, -1) and the slope (frac{5}{3}), we get:

[y - (-1) frac{5}{3}(x - (-1))]

Simplifying the equation:

[y 1 frac{5}{3}x frac{5}{3}]

Step 4: Convert to Slope-Intercept Form

Combine like terms:

[y frac{5}{3}x frac{5}{3} - 1]

Convert the constant -1 to a fraction with the denominator 3:

[y frac{5}{3}x frac{5}{3} - frac{3}{3}]

Simplify the right-hand side:

[y frac{5}{3}x frac{2}{3}]

Conclusion

The equation of the line that passes through point P (-1, -1) and is perpendicular to the line segment joining points A (3, -1) and B (-2, 2) is:

[y frac{5}{3}x frac{2}{3}]

For a visual confirmation, you can graph this equation using an online graphing tool like Desmos. Simply input the equation (y frac{5}{3}x frac{2}{3}) into Desmos and observe the line passing through the point (-1, -1) and perpendicular to the line segment AB.

Additional Practice and Resources

Mastering the concept of finding the equation of a perpendicular line involves practice. If you need further assistance or more detailed explanations, consider exploring additional resources such as textbooks, online tutorials, or educational platforms.

By following the steps outlined in this guide, you will not only solve this specific problem but also enhance your skills in handling similar questions involving lines and slopes.