Finding Equations Of Perpendicular Lines In Point-Slope Form A Step-by-Step Guide

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In mathematics, determining the equation of a line that is perpendicular to another line and passes through a specific point is a fundamental concept in coordinate geometry. This article will guide you through the process of finding such an equation, focusing on the point-slope form. We will use a specific example to illustrate the steps involved, ensuring a clear understanding of the underlying principles.

Understanding Point-Slope Form and Perpendicular Lines

The point-slope form of a linear equation is a powerful tool for representing a line when you know a point on the line and its slope. It is expressed as:

y - y₁ = m(x - x₁)

where:

  • (x₁, y₁) are the coordinates of a known point on the line.
  • m is the slope of the line.

Before we dive into the example, let's discuss the concept of perpendicular lines. Two lines are perpendicular if they intersect at a right angle (90 degrees). A crucial property of perpendicular lines is that their slopes are negative reciprocals of each other. This means if one line has a slope of m, a line perpendicular to it will have a slope of -1/m. This relationship is essential for solving problems involving perpendicular lines.

Determining the Slope of a Perpendicular Line

The first key step in finding the equation of a perpendicular line is to determine its slope. This involves understanding the relationship between the slopes of perpendicular lines, as they are negative reciprocals of each other. The negative reciprocal of a slope is found by inverting the original slope and changing its sign. For example, if a line has a slope of 2, the slope of a line perpendicular to it would be -1/2. Conversely, if a line has a slope of -3/4, the perpendicular slope would be 4/3. This principle ensures that the lines intersect at a right angle, which is the defining characteristic of perpendicularity. In practical terms, calculating the perpendicular slope is a straightforward process, but it's vital to grasp the underlying mathematical reason for this relationship. This understanding helps in visualizing how lines with these slopes will behave on a coordinate plane and why they form a 90-degree angle at their intersection.

Utilizing the Point-Slope Form Effectively

Once the slope of the perpendicular line is determined, the next step is to use the point-slope form to construct the equation of the line. The point-slope form, expressed as y - y₁ = m(x - x₁), is particularly useful when a point on the line and its slope are known. In this equation, m represents the slope, and (x₁, y₁) are the coordinates of the known point. By substituting the calculated perpendicular slope for m and the coordinates of the given point for (x₁, y₁), the equation of the line can be directly formulated. This form is advantageous because it bypasses the need to calculate the y-intercept, which is required in the slope-intercept form (y = mx + b). The point-slope form is not only efficient but also provides a clear representation of the line's characteristics, highlighting both its slope and a specific point it passes through. This method is a cornerstone of coordinate geometry and is essential for solving a variety of problems involving linear equations.

Example: Finding the Equation

Let's consider the problem: Identify an equation in point-slope form for the line perpendicular to y = -1/2x + 11 that passes through the point (4, -8).

  1. Identify the slope of the given line: The equation y = -1/2x + 11 is in slope-intercept form (y = mx + b), where m represents the slope. In this case, the slope of the given line is -1/2.

  2. Determine the slope of the perpendicular line: The slope of a line perpendicular to the given line is the negative reciprocal of -1/2. To find the negative reciprocal, we flip the fraction and change the sign: -1/(-1/2) = 2. So, the slope of the perpendicular line is 2.

  3. Apply the point-slope form: We know the perpendicular line passes through the point (4, -8) and has a slope of 2. Using the point-slope form (y - y₁ = m(x - x₁)), we substitute the values:

    y - (-8) = 2(x - 4)

    Simplifying, we get:

    y + 8 = 2(x - 4)

Verifying the Perpendicularity Condition

To ensure that the lines are indeed perpendicular, it's crucial to verify that their slopes meet the negative reciprocal condition. This verification involves confirming that the product of the slopes of the original line and the new line is -1. If the original line has a slope of -1/2 and the newly derived line has a slope of 2, multiplying these slopes together should result in -1. Mathematically, this is represented as (-1/2) * 2 = -1. This equation confirms that the lines are perpendicular, as the product of their slopes satisfies the condition for perpendicularity. This step is not just a formality but a critical check that validates the entire process, ensuring that the final equation accurately represents a line that is perpendicular to the original one. This thoroughness is essential in mathematics to guarantee the correctness of solutions and to build confidence in the application of geometric principles.

Applying the Point-Slope Form to Diverse Scenarios

The point-slope form is not just a formula but a versatile tool applicable in various scenarios within coordinate geometry and beyond. Its utility extends to problems involving tangent lines in calculus, trajectory calculations in physics, and even in practical applications like determining the path of a vehicle moving at a constant speed and direction. The beauty of the point-slope form lies in its ability to provide a linear equation with minimal information—just a point and a slope. This simplicity makes it an indispensable tool for mathematicians, engineers, and anyone dealing with linear relationships. Understanding how to manipulate and apply this form can significantly enhance problem-solving skills and provide a deeper insight into the behavior of linear functions in various contexts. The more one practices using the point-slope form, the more adept they become at recognizing and solving real-world problems involving linear equations.

Solution

Therefore, the equation in point-slope form for the line perpendicular to y = -1/2x + 11 that passes through (4, -8) is:

y + 8 = 2(x - 4)

So, the correct answer is B.

Conclusion

Finding the equation of a line perpendicular to another line using the point-slope form is a fundamental skill in algebra and geometry. By understanding the relationship between slopes of perpendicular lines and applying the point-slope form, you can confidently solve these types of problems. This article provided a step-by-step guide, illustrating the process with a clear example. Remember to practice and apply these concepts to master them effectively. Mastering these concepts opens doors to more advanced topics in mathematics and its applications in various fields.