Finding The Equation Of A Perpendicular Line A Comprehensive Guide

In mathematics, determining the equation of a line that is perpendicular to a given line and passes through a specific point is a fundamental concept in coordinate geometry. This skill is essential for various applications, including computer graphics, physics simulations, and engineering designs. In this comprehensive guide, we will delve into the step-by-step process of finding the equation of a perpendicular line, ensuring you grasp the underlying principles and can confidently tackle similar problems.

Understanding Perpendicular Lines and Slope-Intercept Form

Before we embark on the problem-solving journey, it's crucial to establish a firm understanding of the key concepts involved. Perpendicular lines are lines that intersect at a right angle (90 degrees). A fundamental property of perpendicular lines is that their slopes are negative reciprocals of each other. This means that if a line has a slope of m, a line perpendicular to it will have a slope of -1/m. This inverse relationship is the cornerstone of our approach.

The slope-intercept form of a linear equation is a standard way of representing a line, expressed as y = mx + b, where m represents the slope of the line and b represents the y-intercept (the point where the line crosses the y-axis). This form provides a clear and concise representation of the line's characteristics, making it easy to visualize and analyze.

Understanding the slope-intercept form is crucial for our task. It allows us to directly identify the slope and y-intercept of a line, which are essential components in determining the equation of a perpendicular line. When we find the equation of the new perpendicular line, we will express it in this form to clearly show its slope and y-intercept.

Step 1: Identify the Slope of the Given Line

The given equation is y = -1/2 x - 5, which is already in slope-intercept form. By comparing this equation to the general form y = mx + b, we can easily identify the slope of the given line. The coefficient of x, which is -1/2, represents the slope. Therefore, the slope of the given line is m = -1/2. This initial step is critical because the slope of the given line is the foundation for determining the slope of the perpendicular line.

Understanding the slope of the given line is like having the first piece of a puzzle. It allows us to use the negative reciprocal relationship to find the slope of the line we are looking for. The negative reciprocal is simply the inverse of the slope with the sign changed. This crucial step connects the given information to the unknown, setting the stage for the next phase of our calculation.

We now know the slope of the original line, but to find the perpendicular line, we need its slope. This is where the concept of negative reciprocals comes into play. By taking the negative reciprocal of -1/2, we will find the slope of the line perpendicular to the given line. This slope will be used in the equation of the new line, along with the given point, to fully define the perpendicular line.

Step 2: Calculate the Slope of the Perpendicular Line

Now, we apply the concept of negative reciprocals to find the slope of the line perpendicular to the given line. As mentioned earlier, the slope of a line perpendicular to a line with slope m is -1/m. In our case, m = -1/2. So, the slope of the perpendicular line is -1/(-1/2), which simplifies to 2. Therefore, the slope of the perpendicular line is m = 2. This calculation is a pivotal step, as it provides the slope that will define the direction of our perpendicular line.

Finding the slope of the perpendicular line involves a simple yet crucial mathematical operation. Taking the negative reciprocal might seem like a minor step, but it fundamentally changes the direction of the line. This new slope ensures that the line we construct will indeed be perpendicular to the original line, meeting the core requirement of the problem. This step is like turning a key, unlocking the possibility of finding the correct equation.

With the slope of the perpendicular line in hand, we are one step closer to our final answer. This slope, combined with the point that the line passes through, will allow us to use the point-slope form of a linear equation to construct the equation of the perpendicular line. Understanding how slopes relate to perpendicular lines is not just about memorizing a formula; it's about grasping the geometric relationship between lines in space.

Step 3: Use the Point-Slope Form

The point-slope form of a linear equation is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. We know the perpendicular line passes through the point (2, 7), so x1 = 2 and y1 = 7. We also know that the slope of the perpendicular line is m = 2. Substituting these values into the point-slope form, we get y - 7 = 2(x - 2). This equation represents the perpendicular line in point-slope form. This step is a bridge connecting the slope and the given point, leading us to the equation of the line.

The point-slope form is a powerful tool because it allows us to construct the equation of a line using just a single point and the slope. It's a flexible form that can easily be converted into other forms, such as the slope-intercept form. The point-slope form is like a blueprint, providing the necessary information to draw the line on a coordinate plane. It's a tangible representation of the line's characteristics, making it easier to visualize and manipulate.

By plugging in the values into the point-slope form, we are essentially anchoring the line to a specific point while giving it a specific direction. The slope dictates the line's steepness, and the point ensures that the line passes through the correct location on the coordinate plane. This form is a versatile stepping stone in finding the equation of a line, and it perfectly suits our needs in this problem.

Step 4: Convert to Slope-Intercept Form

To express the equation in slope-intercept form (y = mx + b), we need to solve the equation y - 7 = 2(x - 2) for y. First, distribute the 2 on the right side of the equation: y - 7 = 2x - 4. Then, add 7 to both sides of the equation to isolate y: y = 2x - 4 + 7. Finally, simplify the equation: y = 2x + 3. This is the equation of the perpendicular line in slope-intercept form. This final step transforms the equation into a familiar and easily interpretable form.

Converting to slope-intercept form is like putting the finishing touches on a masterpiece. It presents the equation in a clear and concise manner, highlighting the slope and y-intercept. This form makes it easy to visualize the line on a graph and compare it to other lines. The slope-intercept form is a universal language for lines, making it easy to communicate and understand their properties.

The process of converting from point-slope form to slope-intercept form involves algebraic manipulation, which is a fundamental skill in mathematics. By following the steps of distribution and simplification, we arrive at the final equation that represents the perpendicular line. This conversion is not just about finding the answer; it's about demonstrating our understanding of the relationships between different forms of linear equations.

Final Answer

The equation of the line perpendicular to y = -1/2 x - 5 that passes through the point (2, 7) in slope-intercept form is y = 2x + 3**. This equation clearly shows the slope of the perpendicular line (2) and its y-intercept (3). We have successfully navigated through the steps, from understanding perpendicular lines to expressing the final answer in the desired format.

This final answer is the culmination of our efforts, a precise and elegant representation of the perpendicular line. It encapsulates all the key concepts and calculations we have discussed, providing a definitive solution to the problem. This equation is not just a set of symbols; it's a powerful tool that allows us to analyze, predict, and manipulate lines in the coordinate plane.

In conclusion, finding the equation of a perpendicular line involves understanding the relationship between slopes, using the point-slope form, and converting to slope-intercept form. By mastering these steps, you can confidently solve a wide range of problems in coordinate geometry and beyond. This ability to find perpendicular lines is a cornerstone of mathematical thinking, applicable in various fields and disciplines.