In coordinate geometry, a fundamental concept is the perpendicular bisector of a line segment. This is a line that not only intersects the given line segment at its midpoint but also forms a right angle with it. Understanding how to find the equation of a perpendicular bisector is crucial in various mathematical applications, from geometric proofs to coordinate plane problems. In this article, we will delve into the process of determining the equation of a perpendicular bisector in slope-intercept form, given the midpoint of a line segment. This topic is a cornerstone of high school geometry and algebra, often appearing in standardized tests and advanced mathematical studies. We will explore the underlying principles, the step-by-step methodology, and the significance of this concept in a comprehensive manner, ensuring that you grasp not just the how but also the why behind each step. This article aims to provide a clear, concise, and thorough explanation, empowering you to tackle similar problems with confidence and proficiency. To begin, let's clarify the core concepts: a line segment, its midpoint, and what constitutes a perpendicular bisector. A line segment is a part of a line that is bounded by two distinct endpoints, and its midpoint is the point that divides the segment into two equal parts. The perpendicular bisector, as the name suggests, performs two key functions: it bisects (divides into two equal parts) the line segment and is perpendicular (at a 90-degree angle) to it. The slope-intercept form of a line, represented as y = mx + b, is a particularly useful form because it explicitly shows the slope (m) and the y-intercept (b) of the line. This form makes it easy to visualize and analyze the line's characteristics on a coordinate plane. The slope, m, indicates the steepness and direction of the line, while the y-intercept, b, is the point where the line crosses the y-axis. With these foundational concepts in place, we can proceed to the methodology of finding the perpendicular bisector's equation.
Understanding the Problem
Let's restate the problem clearly: Given a line segment with a midpoint at (-1, -2), our objective is to determine the equation of the line that is both perpendicular to this line segment and passes through its midpoint. The equation should be expressed in slope-intercept form, which is y = mx + b, where m represents the slope and b represents the y-intercept. Before we dive into the solution, it's crucial to understand the geometric properties that underpin this problem. First, a bisector, by definition, cuts a line segment into two equal parts. In this context, the perpendicular bisector passes through the midpoint of the given line segment, ensuring that it divides the segment equally. Second, the term "perpendicular" signifies that the bisector intersects the line segment at a right angle (90 degrees). This perpendicularity introduces a special relationship between the slopes of the two lines. Specifically, if a line has a slope of m, then any line perpendicular to it will have a slope of -1/m. This is a fundamental principle we will use to find the slope of the perpendicular bisector. Now, let's outline the steps we will take to solve this problem. 1. Determine the slope of the original line segment: While we are not given the endpoints of the line segment directly, we can infer that we would need this information if we had it. However, since we are not provided with the endpoints, we will proceed without this information for now and see how we can still solve the problem using the properties of perpendicular bisectors. 2. Calculate the slope of the perpendicular bisector: Using the principle that the slopes of perpendicular lines are negative reciprocals of each other, we will find the slope of the perpendicular bisector. 3. Use the midpoint and the slope to find the equation of the perpendicular bisector: We know that the perpendicular bisector passes through the midpoint (-1, -2). We will use this point and the slope calculated in the previous step to determine the equation of the line in slope-intercept form. This involves substituting the known values into the equation y = mx + b and solving for b. By following these steps, we will arrive at the equation of the perpendicular bisector in the desired form. Let's proceed with the first step, keeping in mind the information we have and how we can use it effectively.
Steps to Find the Perpendicular Bisector
Step 1: Finding the Slope of the Perpendicular Bisector
As previously mentioned, the slope of the original line segment is crucial for finding the slope of its perpendicular bisector. However, in this problem, we are not explicitly given the endpoints of the line segment. This might seem like a roadblock, but it's a common scenario in mathematics where we need to think creatively and use the information we have in a clever way. Since we don't have the endpoints, we cannot directly calculate the slope using the traditional formula (m = (y2 - y1) / (x2 - x1)). Instead, let's denote the slope of the original line segment as m. The key principle here is that the slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. This means if the original line has a slope of m, the perpendicular bisector will have a slope of -1/m. This relationship is derived from the fact that the product of the slopes of two perpendicular lines is -1. Mathematically, this can be expressed as: m * (-1/m) = -1. While we don't know the exact value of m, we can still express the slope of the perpendicular bisector in terms of m. This might seem abstract, but it's a necessary step to proceed with the problem. We'll keep this in mind and move on to the next step, where we'll see how we can use the midpoint information to further refine our understanding and eventually find the equation of the perpendicular bisector. It's important to remember that mathematical problem-solving often involves working with unknowns and using relationships between variables to gradually narrow down the possibilities. This is precisely what we are doing here. By acknowledging that we don't have all the information initially but understanding the fundamental principles of perpendicular lines, we can set the stage for the subsequent steps in the solution. Let's proceed to the next step and see how the midpoint comes into play.
Step 2: Using the Midpoint and Slope-Intercept Form
Now that we have the slope of the perpendicular bisector expressed as -1/m, the next crucial step is to utilize the given midpoint (-1, -2). The midpoint provides a specific point that lies on the perpendicular bisector. This is vital information because we can use it, along with the slope, to determine the equation of the line in slope-intercept form (y = mx + b). Recall that the slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. We already have an expression for the slope of the perpendicular bisector (-1/m), and we have a point (-1, -2) that lies on this line. We can substitute these values into the slope-intercept equation to solve for the y-intercept, b. Substituting x = -1, y = -2, and m = -1/m into the equation y = mx + b, we get: -2 = (-1/m) * (-1) + b. This simplifies to: -2 = 1/m + b. Now, we need to solve for b. To isolate b, we subtract 1/m from both sides of the equation: b = -2 - 1/m. This gives us an expression for the y-intercept, b, in terms of m. At this point, you might notice that we still have an unknown (m) in our expression for b. This is where the problem becomes a bit more intricate, and we need to look back at the given options to see if we can deduce the value of m. Alternatively, if we had the slope of the original line segment (the m we initially referred to), we could substitute it into our expressions for the slope of the perpendicular bisector and the y-intercept, thus fully defining the equation of the perpendicular bisector. However, without the original slope or additional information about the line segment, we must rely on the provided answer choices to determine the correct equation. The key here is to analyze how the slope and y-intercept are related in our derived expressions and see which of the given options fits this relationship. By carefully comparing the slopes and y-intercepts of the answer choices with our derived expressions, we can identify the equation that represents the perpendicular bisector. This step highlights the importance of connecting the theoretical understanding of perpendicular bisectors and slope-intercept form with practical problem-solving techniques. Let's now proceed to analyze the answer choices and see which one aligns with our findings.
Analyzing the Answer Choices
Now, let's examine the given answer choices. We have three potential equations for the perpendicular bisector: 1. y = -4x - 4 2. y = -4x - 6 3. y = (1/4)x - 4 Our goal is to determine which of these equations correctly represents the perpendicular bisector of the line segment, given that the midpoint is (-1, -2). We've already established that the slope of the perpendicular bisector is -1/m and the y-intercept is b = -2 - 1/m, where m is the slope of the original line segment. Let's analyze each option in terms of its slope and y-intercept and see if it aligns with these conditions.
Option 1: y = -4x - 4
In this equation, the slope is -4 and the y-intercept is -4. If this were the correct equation, it would imply that -1/m = -4. Solving for m, we get m = 1/4. Now, let's check if the y-intercept also fits the condition b = -2 - 1/m. Substituting m = 1/4 into this equation, we get b = -2 - 1/(1/4) = -2 - 4 = -6. However, the y-intercept in the equation is -4, not -6. Therefore, this option does not satisfy both conditions and is likely incorrect.
Option 2: y = -4x - 6
In this equation, the slope is -4 and the y-intercept is -6. As we calculated in the analysis of Option 1, if the slope of the perpendicular bisector is -4, then the slope of the original line segment would be m = 1/4. We also found that with m = 1/4, the y-intercept of the perpendicular bisector should be b = -6. This option matches both the slope and y-intercept conditions we derived. Therefore, this equation is a strong contender for the correct answer.
Option 3: y = (1/4)x - 4
In this equation, the slope is 1/4 and the y-intercept is -4. If this were the correct equation, then -1/m = 1/4, which implies m = -4. Substituting m = -4 into the y-intercept condition b = -2 - 1/m, we get b = -2 - 1/(-4) = -2 + 1/4 = -7/4. However, the y-intercept in this equation is -4, not -7/4. Therefore, this option does not satisfy both conditions and is likely incorrect. Based on our analysis, Option 2 (y = -4x - 6) is the only equation that satisfies both the slope and y-intercept conditions we derived from the given midpoint and the properties of perpendicular bisectors. Therefore, we can confidently conclude that this is the correct equation for the perpendicular bisector.
Final Answer
After carefully analyzing the problem and the provided answer choices, we have determined that the equation of the perpendicular bisector of the given line segment, with a midpoint at (-1, -2), is: y = -4x - 6 This conclusion was reached by: 1. Understanding the properties of perpendicular bisectors, specifically the relationship between the slopes of perpendicular lines and the fact that a bisector passes through the midpoint of a line segment. 2. Expressing the slope of the perpendicular bisector in terms of the slope of the original line segment (-1/m). 3. Using the midpoint coordinates and the slope-intercept form of a line to derive an expression for the y-intercept of the perpendicular bisector (b = -2 - 1/m). 4. Analyzing each answer choice by comparing its slope and y-intercept to the derived conditions, ultimately identifying Option 2 (y = -4x - 6) as the only equation that satisfies both conditions. This problem demonstrates the importance of understanding fundamental geometric principles and how to apply them in a coordinate geometry context. It also highlights the value of a systematic approach to problem-solving, where we break down the problem into smaller, manageable steps and use the given information strategically. In summary, the equation y = -4x - 6 accurately represents the perpendicular bisector of the line segment with a midpoint at (-1, -2).