What is the Method for Finding a Perpendicular Line to Another Line Passing Through a Specific Point?

In geometry, finding a line that is perpendicular to another line at a specific point is a fundamental task. This can be achieved without the need for matrices or determinants, making it accessible for anyone with basic algebraic skills. Let's explore the method in detail.

Understanding the Basics

First, let's recall the equation of a line. A line with slope m passing through a point (x_0, y_0) can be written as:

y m(x - x_0) y_0

For simplicity, let's rewrite this as:

y mx (y_0 - mx_0)

How to Find the Perpendicular Line

Let's consider a specific example. We have a line with a known slope and a point through which we need to pass a perpendicular line.

Example Problem

Suppose we have the line y 3x - 2 and we need to find the equation of the perpendicular line that passes through the point (2, 3).

Step-by-Step Solution

1. **Identify the slope of the original line**: The given line is in the form y 3x - 2. Therefore, the slope m of this line is 3.

2. **Determine the slope of the perpendicular line**: The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Hence, the slope of the perpendicular line is:

m_{perp} -frac{1}{3}

3. **Use the point-slope form to find the equation of the perpendicular line**: The point-slope form of a line is y - y_1 m(x - x_1). Using the point (2, 3) and the slope m_{perp} -frac{1}{3}, we write:

y - 3 -frac{1}{3}(x - 2)

4. **Simplify the equation**:

y - 3 -frac{1}{3}x frac{2}{3}

y -frac{1}{3}x frac{2}{3} 3

y -frac{1}{3}x frac{11}{3}

General Method

Consider a general line given by y ax b. We need to find the equation of the perpendicular line that passes through the point (x_0, y_0).

Step-by-Step General Method

Identify the slope of the original line: a.

Determine the slope of the perpendicular line: -frac{1}{a}.

Use the point-slope form: y - y_0 -frac{1}{a}(x - x_0).

Simplify to get the equation of the perpendicular line.

Example: y ax b

Suppose we have the line y 5x 10 and we want to find the equation of the perpendicular line passing through the point (4, 7).

1. **Identify the slope of the original line**: a 5.

2. **Determine the slope of the perpendicular line**: -frac{1}{5}.

3. **Use the point-slope form:

y - 7 -frac{1}{5}(x - 4)

4. **Simplify:

y - 7 -frac{1}{5}x frac{4}{5}

y -frac{1}{5}x frac{4}{5} 7

y -frac{1}{5}x frac{39}{5}

Thus, the equation of the perpendicular line is:

y -frac{1}{5}x frac{39}{5}

Conclusion

By understanding the relationship between the slopes of perpendicular lines and the point-slope form, we can easily find the equation of a perpendicular line that passes through a specific point. The process is straightforward and does not require complex mathematical tools like matrices or determinants.