Finding The Equation Of A Perpendicular Line A Step-by-Step Guide

In the realm of mathematics, particularly in coordinate geometry, the concept of perpendicular lines holds significant importance. Understanding the properties of perpendicular lines is crucial for solving various problems, from finding equations of lines to analyzing geometric shapes. This article delves into the intricacies of determining the equation of a line perpendicular to a given line and passing through a specific point. We will explore the underlying principles, step-by-step solutions, and practical applications of this concept.

Understanding Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle (90 degrees). The relationship between their slopes is fundamental to solving problems involving perpendicularity. The slope of a line, often denoted as 'm', represents its steepness or inclination. For two lines to be perpendicular, the product of their slopes must be -1. This means that if one line has a slope of 'm', the slope of a line perpendicular to it will be '-1/m'.

In the context of coordinate geometry, a horizontal line has a slope of 0, and a vertical line has an undefined slope. A line perpendicular to a horizontal line is always a vertical line, and vice versa. This special relationship simplifies the process of finding perpendicular lines in certain cases.

The equation of a line can be expressed in various forms, including slope-intercept form (y = mx + b), point-slope form (y - y1 = m(x - x1)), and standard form (Ax + By = C). The choice of form depends on the given information and the desired outcome. In this article, we will primarily focus on the slope-intercept and point-slope forms, as they are particularly useful for finding the equation of a line given its slope and a point it passes through.

Problem Statement

Let's consider the problem of finding the equation of a line that is perpendicular to the line $y = \frac{1}{4}$ and passes through the point (-6, -9). This problem exemplifies the application of the principles discussed above. We need to determine the slope of the perpendicular line and then use the given point to find the equation of the line.

To solve this problem, we will follow a step-by-step approach:

1. Identify the slope of the given line.
2. Determine the slope of the perpendicular line.
3. Use the point-slope form to find the equation of the perpendicular line.
4. Simplify the equation to the desired form.

By carefully following these steps, we can arrive at the correct equation of the perpendicular line.

Step 1: Identify the Slope of the Given Line

The given line is represented by the equation $y = \frac{1}{4}$. This equation is in slope-intercept form (y = mx + b), where 'm' represents the slope and 'b' represents the y-intercept. In this case, the equation can be rewritten as y = 0x + (1/4). This clearly shows that the slope (m) of the given line is 0. A line with a slope of 0 is a horizontal line.

The slope of a line is a measure of its steepness and direction. A horizontal line has no steepness, hence its slope is 0. Vertical lines, on the other hand, have an undefined slope because their steepness is infinite. Understanding the relationship between the slope and the direction of a line is crucial for solving problems involving perpendicular lines.

Identifying the slope of the given line is the first step towards finding the equation of the perpendicular line. Once we know the slope of the given line, we can determine the slope of the perpendicular line using the relationship between their slopes.

Step 2: Determine the Slope of the Perpendicular Line

Since the given line has a slope of 0 (it is a horizontal line), a line perpendicular to it must be a vertical line. Vertical lines have an undefined slope. This is because the change in x is zero, and division by zero is undefined.

The relationship between the slopes of perpendicular lines is that their product is -1. However, this relationship doesn't directly apply when one of the lines is horizontal or vertical, as the slope of a vertical line is undefined. Instead, we rely on the geometric understanding that a line perpendicular to a horizontal line is always a vertical line.

The slope of the perpendicular line is a crucial piece of information for finding its equation. In this case, knowing that the perpendicular line is vertical simplifies the process, as we know its equation will be of the form x = constant.

Step 3: Use the Point-Slope Form (or Vertical Line Form) to Find the Equation

Since we know the perpendicular line is vertical and passes through the point (-6, -9), its equation must be of the form x = constant. The constant value is the x-coordinate of the point the line passes through. Therefore, the equation of the perpendicular line is x = -6.

The point-slope form of a line (y - y1 = m(x - x1)) is generally used when we know the slope of the line and a point it passes through. However, in this case, since the slope is undefined, we cannot directly apply the point-slope form. Instead, we rely on the understanding that a vertical line has a constant x-value, and the equation is simply x equals that constant.

The equation x = -6 represents a vertical line that passes through all points with an x-coordinate of -6. This includes the point (-6, -9), as required by the problem statement.

Step 4: Verify the Solution

To ensure our solution is correct, we can verify that the line x = -6 is indeed perpendicular to the line y = 1/4 and passes through the point (-6, -9). The line x = -6 is a vertical line, and the line y = 1/4 is a horizontal line. Vertical and horizontal lines are always perpendicular to each other. Additionally, the point (-6, -9) lies on the line x = -6 because its x-coordinate is -6.

By verifying our solution, we can have confidence that we have correctly determined the equation of the perpendicular line. This step is crucial in problem-solving to avoid errors and ensure accuracy.

Options Analysis

Now, let's analyze the given options in the problem:

A. x = -9 B. x = -6 C. y = -9 D. y = -4

We have determined that the equation of the line perpendicular to y = 1/4 and passing through (-6, -9) is x = -6. Comparing this to the options, we can see that option B (x = -6) is the correct answer.

The other options are incorrect because:

• Option A (x = -9) represents a vertical line, but it does not pass through the point (-6, -9).
• Option C (y = -9) represents a horizontal line, which is not perpendicular to y = 1/4.
• Option D (y = -4) represents a horizontal line, which is also not perpendicular to y = 1/4.

By carefully analyzing the options and comparing them to our solution, we can confidently select the correct answer.

Conclusion

In conclusion, the equation of the line perpendicular to $y = \frac{1}{4}$ and passing through the point (-6, -9) is x = -6. This solution was obtained by understanding the relationship between the slopes of perpendicular lines, recognizing that a line perpendicular to a horizontal line is vertical, and using the given point to determine the equation of the vertical line.

This problem highlights the importance of understanding fundamental concepts in coordinate geometry, such as slopes, perpendicularity, and the equations of lines. By mastering these concepts, one can effectively solve a wide range of problems involving lines and their relationships.

The step-by-step approach outlined in this article can be applied to similar problems involving perpendicular lines. By carefully identifying the slope of the given line, determining the slope of the perpendicular line, and using the appropriate form to find the equation, one can confidently arrive at the correct solution. Remember to always verify your solution to ensure accuracy and avoid errors.

Understanding the characteristics of perpendicular lines and their equations is a fundamental concept in mathematics, especially in coordinate geometry. This concept allows us to solve a variety of problems, including finding equations of lines, understanding geometric shapes, and their relationships. In this article, we explored how to determine the equation of a line perpendicular to a given line, passing through a specified point. We went through the problem-solving process step by step, starting with understanding the properties of perpendicular lines, identifying the slope of the given line, determining the slope of the perpendicular line, and finally using the point-slope form to arrive at the correct equation.

Our problem involved finding a line perpendicular to $y = \frac{1}{4}$ that passes through the point (-6, -9). We identified that the given line is a horizontal line with a slope of 0. A line perpendicular to a horizontal line is a vertical line, which has an undefined slope. We then recognized that the equation of a vertical line passing through a point (x, y) is simply x = the x-coordinate of the point. Thus, the equation of the line perpendicular to the given line and passing through the specified point was determined to be x = -6.

The importance of understanding slopes cannot be overstated. The slope of a line tells us how steeply the line rises or falls as we move from left to right. A horizontal line has a slope of 0, meaning it neither rises nor falls. A vertical line, on the other hand, has an undefined slope because it rises (or falls) infinitely for no change in the horizontal direction. The relationship between slopes of perpendicular lines is crucial: if a line has a slope m, then a line perpendicular to it has a slope of -1/m. However, in the case of horizontal and vertical lines, this relationship manifests differently; a line perpendicular to a horizontal line is vertical, and vice versa.

Using the point-slope form of a line, which is generally given as $y - y_1 = m(x - x_1)$, is a useful method when you have a point on the line and the slope. However, for vertical lines, this form is not directly applicable because the slope is undefined. Instead, we use the fact that all points on a vertical line have the same x-coordinate. Therefore, the equation of a vertical line is simply x = constant, where the constant is the x-coordinate of any point on the line.

Verifying the solution is a critical step in problem-solving. We confirmed that x = -6 is indeed perpendicular to $y = \frac{1}{4}$ because vertical and horizontal lines are always perpendicular. We also verified that the point (-6, -9) lies on the line x = -6 because the x-coordinate of the point matches the constant value in the equation. This step ensures that we have not made any mistakes in our calculations or reasoning.

In the options analysis, we carefully examined each choice to see which one matched our solution. We determined that option B, x = -6, was the correct answer. The other options were either not vertical lines (and thus not perpendicular to the horizontal line) or did not pass through the given point.

In summary, solving problems involving perpendicular lines requires a solid understanding of slopes, equations of lines, and geometric relationships. This article has provided a detailed explanation of how to approach such problems, including identifying the key concepts, applying the appropriate formulas, and verifying the solution. The ability to solve these types of problems is crucial in various fields, including mathematics, physics, engineering, and computer graphics. By mastering these concepts, you will be better equipped to tackle more advanced problems and applications in these areas.

Mastering the concept of finding perpendicular lines is a cornerstone in mathematics, particularly within the domain of coordinate geometry. It's a skill that extends beyond the classroom, finding its application in diverse fields like engineering, physics, computer graphics, and architecture. This article has dissected the process of determining the equation of a line that stands perpendicular to a given line and traverses a specific point. Our approach has been meticulous, breaking down the problem into manageable steps, each grounded in fundamental principles.

Our illustrative problem presented the challenge of identifying the line perpendicular to $y = \frac{1}{4}$ and simultaneously passing through the coordinate point (-6, -9). This scenario encapsulates the essence of coordinate geometry – bridging algebraic equations with geometric interpretations. The initial step in our journey involved recognizing the nature of the given line. The equation $y = \frac{1}{4}$ represents a horizontal line, a visual cue that simplifies our quest for its perpendicular counterpart.

The slope of a line is an indispensable concept, serving as a numeric descriptor of a line's inclination or steepness. A horizontal line, by its very definition, exhibits a slope of 0, signifying no vertical change for any horizontal movement. Conversely, a vertical line presents an undefined slope, a consequence of an infinite vertical change over zero horizontal change. This distinction is crucial when navigating problems involving perpendicularity.

Perpendicular lines intersect at a precise right angle (90 degrees), a geometric relationship that translates to an algebraic condition involving their slopes. The general rule dictates that the product of the slopes of two perpendicular lines is -1. However, this rule requires a nuanced understanding when dealing with horizontal and vertical lines. A line perpendicular to a horizontal line is invariably vertical, and vice versa. This geometric certainty streamlines our problem-solving process.

The point-slope form, a versatile tool in the arsenal of coordinate geometry, is expressed as $y - y_1 = m(x - x_1)$, where 'm' denotes the slope and $(x_1, y_1)$ represents a point on the line. This form proves invaluable when the slope and a point are known. However, when confronted with a vertical line, where the slope is undefined, we pivot to the intrinsic property of vertical lines – a constant x-coordinate. The equation of a vertical line, therefore, takes the form x = constant.

In our specific problem, the perpendicular line must traverse the point (-6, -9). Consequently, the x-coordinate of this point, -6, dictates the equation of our perpendicular line. Thus, the equation is x = -6, a vertical line slicing through all points with an x-coordinate of -6.

Verification of our solution is a non-negotiable step, a safeguard against errors. We confirmed the perpendicularity of x = -6 and $y = \frac{1}{4}$ based on the inherent perpendicularity of vertical and horizontal lines. Furthermore, we validated that the point (-6, -9) indeed resides on the line x = -6, solidifying the accuracy of our solution.

The multiple-choice options presented a spectrum of possibilities, each demanding careful scrutiny. Option B, x = -6, emerged as the victor, aligning perfectly with our derived solution. The other options faltered, either failing to represent vertical lines or missing the crucial point (-6, -9).

In summation, the journey to find the equation of a perpendicular line is a testament to the interconnectedness of algebraic equations and geometric principles. A firm grasp of slopes, equations of lines, and the geometric dance of perpendicularity forms the bedrock of this skill. This article has illuminated a structured approach to such problems, emphasizing the importance of conceptual understanding, strategic application of formulas, and the unwavering commitment to solution verification. The ability to navigate these problems is not merely an academic exercise; it's a gateway to a deeper appreciation of mathematics and its pervasive applications in the world around us.