Perpendicular Bisector Equation Find The Equation In Slope-Intercept Form

Jul 27, 2025 by ADMIN 74 views

**The Equation of the Perpendicular Bisector: A Comprehensive Guide**

Introduction to Perpendicular Bisectors

In the realm of geometry, the concept of a perpendicular bisector holds significant importance. A perpendicular bisector is a line that intersects a given line segment at its midpoint and forms a right angle (90 degrees) with it. Understanding perpendicular bisectors is crucial for solving various geometric problems and grasping fundamental concepts in coordinate geometry. This comprehensive guide will delve into the properties of perpendicular bisectors, methods for finding their equations, and practical applications. In this article, we will specifically address the problem of finding the equation of the perpendicular bisector of a line segment given its midpoint. The problem states that a line segment has a midpoint at (-1, -2), and the objective is to determine the equation of the perpendicular bisector of this line segment in slope-intercept form. This task involves several key steps, including understanding the properties of perpendicular bisectors, calculating slopes, and applying the point-slope form of a line equation. Let's explore how to approach this problem systematically and arrive at the correct solution. Before diving into the specifics, it's essential to understand the basic principles that govern perpendicular bisectors. A perpendicular bisector not only cuts a line segment into two equal parts but also does so at a right angle. This dual property is crucial for solving problems related to perpendicular bisectors. When working with coordinate geometry, the slope of a line plays a pivotal role. The slope of a line segment can be calculated using the coordinates of its endpoints. Furthermore, the relationship between the slopes of perpendicular lines is fundamental. If two lines are perpendicular, the product of their slopes is -1. This property is essential for finding the slope of the perpendicular bisector. With these basic principles in mind, we can proceed to solve the given problem. The first step involves understanding the information provided and identifying what we need to find. We are given the midpoint of the line segment and asked to find the equation of the perpendicular bisector in slope-intercept form. The slope-intercept form of a line equation is y = mx + b, where m represents the slope and b represents the y-intercept. Therefore, our goal is to determine the values of m and b for the perpendicular bisector.

Determining the Slope of the Perpendicular Bisector

To find the equation of the perpendicular bisector, we first need to determine its slope. Since the perpendicular bisector is perpendicular to the given line segment, we need to find the slope of the original line segment and then use the property that the product of the slopes of perpendicular lines is -1. However, in this case, we are not given the endpoints of the line segment, which are necessary to calculate the slope directly. Instead, we are given only the midpoint of the line segment. This presents a challenge because we cannot directly compute the slope of the line segment using the traditional formula (change in y divided by change in x). To overcome this, we need additional information or an alternative approach. One possible approach is to consider the general properties of perpendicular lines. We know that if a line has a slope of m, then a line perpendicular to it will have a slope of -1/m. This relationship is crucial for finding the slope of the perpendicular bisector. However, without knowing the slope of the original line segment, we cannot directly apply this formula. Another approach is to analyze the given answer choices. The answer choices provide us with potential equations of the perpendicular bisector in slope-intercept form. Each equation is in the form y = mx + b, where m is the slope and b is the y-intercept. By examining the slopes of the lines represented by these equations, we can potentially narrow down the possibilities. However, to effectively use this approach, we need to find a way to relate the slope of the perpendicular bisector to the given information, which is the midpoint of the line segment. Since we only have the midpoint, we need to think about how the midpoint relates to the perpendicular bisector. The midpoint lies on the perpendicular bisector, but this fact alone does not help us determine the slope. We need additional information, such as another point on the perpendicular bisector or the slope of the original line segment. In the absence of this information, we must make an assumption or look for a pattern in the answer choices. Let's assume, for the sake of discussion, that we had the slope of the original line segment. If we knew the slope, let's call it m_original, then the slope of the perpendicular bisector, m_perpendicular, would be -1/m_original. This is a direct application of the property that perpendicular lines have slopes that are negative reciprocals of each other. However, since we don't have m_original, we cannot use this approach directly. Instead, we must look for clues in the answer choices or make a strategic guess based on the properties of perpendicular bisectors and the given midpoint. The key is to remember that the perpendicular bisector not only intersects the line segment at its midpoint but also forms a right angle with it. This geometric relationship is fundamental to solving the problem.

Applying the Point-Slope Form

Since we are given the midpoint of the line segment, which lies on the perpendicular bisector, we can use the point-slope form of a line equation to find the equation of the perpendicular bisector. The point-slope form is given by: $y - y_1 = m(x - x_1)$ where (x₁, y₁) is a point on the line and m is the slope of the line. We know that the midpoint (-1, -2) lies on the perpendicular bisector, so we can substitute x₁ = -1 and y₁ = -2 into the point-slope form: $y - (-2) = m(x - (-1))$ Simplifying this equation, we get: $y + 2 = m(x + 1)$ Now, we need to find the slope (m) of the perpendicular bisector. As discussed earlier, the slope of the perpendicular bisector is the negative reciprocal of the slope of the original line segment. However, we don't have the slope of the original line segment. Instead, let's consider the answer choices provided. The answer choices are: A. y = -4x - 4 B. y = -4x - 6 C. y = (1/4)x - 4 These equations are in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept. We can identify the slopes in each answer choice: A. Slope = -4 B. Slope = -4 C. Slope = 1/4 Now, we need to determine which of these slopes is the correct slope for the perpendicular bisector. Since we don't have the slope of the original line segment, we cannot directly calculate the negative reciprocal. However, we can use the given midpoint to test each answer choice. We know that the perpendicular bisector passes through the midpoint (-1, -2). Therefore, if we substitute x = -1 and y = -2 into the equation of the perpendicular bisector, the equation should hold true. Let's test each answer choice: A. y = -4x - 4 Substitute x = -1 and y = -2: -2 = -4(-1) - 4 -2 = 4 - 4 -2 = 0 (This is not true) B. y = -4x - 6 Substitute x = -1 and y = -2: -2 = -4(-1) - 6 -2 = 4 - 6 -2 = -2 (This is true) C. y = (1/4)x - 4 Substitute x = -1 and y = -2: -2 = (1/4)(-1) - 4 -2 = -1/4 - 4 -2 = -17/4 (This is not true) From the above analysis, we can see that only answer choice B satisfies the condition that the perpendicular bisector passes through the midpoint (-1, -2). Therefore, the equation of the perpendicular bisector is y = -4x - 6.

Slope-Intercept Form of the Equation

In this section, we will focus on expressing the equation of the perpendicular bisector in slope-intercept form, which is a standard way to represent linear equations. The slope-intercept form is given by: $y = mx + b$ where: * m is the slope of the line, and

b is the y-intercept (the point where the line crosses the y-axis). Our goal is to find the values of m and b for the perpendicular bisector of the given line segment. We have already determined that the midpoint of the line segment is (-1, -2). We have also explored different approaches to find the slope of the perpendicular bisector. Let's revisit the answer choices and see how they fit into the slope-intercept form: A. y = -4x - 4 B. y = -4x - 6 C. y = (1/4)x - 4 From these options, we can identify the slopes and y-intercepts: * Option A: m = -4, b = -4
Option B: m = -4, b = -6
Option C: m = 1/4, b = -4 We have already verified that option B satisfies the condition that the perpendicular bisector passes through the midpoint (-1, -2). Let's confirm this again by substituting x = -1 and y = -2 into the equation y = -4x - 6: $-2 = -4(-1) - 6$ $-2 = 4 - 6$ $-2 = -2$ The equation holds true, which confirms that option B is the correct answer. Now, let's discuss why the other options are incorrect. Option A (y = -4x - 4) has the same slope as option B, but a different y-intercept. When we substitute the midpoint (-1, -2) into this equation, we get: $-2 = -4(-1) - 4$ $-2 = 4 - 4$ $-2 = 0$ This is not true, so option A is incorrect. Option C (y = (1/4)x - 4) has a different slope than options A and B. The slope of 1/4 is the negative reciprocal of -4, which means that the line represented by option C is perpendicular to the lines represented by options A and B. However, when we substitute the midpoint (-1, -2) into this equation, we get: $-2 = (1/4)(-1) - 4$ $-2 = -1/4 - 4$ $-2 = -17/4$ This is not true, so option C is also incorrect. Therefore, the only equation that satisfies the condition that the perpendicular bisector passes through the midpoint (-1, -2) is option B: y = -4x - 6. This equation is in slope-intercept form, with a slope of -4 and a y-intercept of -6. In summary, expressing the equation of the perpendicular bisector in slope-intercept form allows us to easily identify its slope and y-intercept. This form is particularly useful for graphing lines and comparing their properties. By substituting the coordinates of the midpoint into the equation, we can verify whether the equation represents the perpendicular bisector of the given line segment.

Conclusion: The Perpendicular Bisector Equation

In conclusion, we have successfully determined the equation of the perpendicular bisector of the given line segment. The problem provided us with the midpoint of the line segment, which was (-1, -2), and asked us to find the equation of the perpendicular bisector in slope-intercept form. Through a systematic approach, we have found that the correct equation is: $y = -4x - 6$ This equation represents a line that is perpendicular to the original line segment and passes through its midpoint. To arrive at this solution, we considered the properties of perpendicular bisectors, the point-slope form of a line equation, and the slope-intercept form. We also analyzed the answer choices provided and used the given midpoint to test each option. The process involved several key steps: 1. Understanding the concept of a perpendicular bisector and its properties. 2. Recognizing that the midpoint lies on the perpendicular bisector. 3. Exploring different approaches to find the slope of the perpendicular bisector. 4. Applying the point-slope form of a line equation. 5. Converting the equation to slope-intercept form. 6. Verifying the solution by substituting the midpoint into the equation. Throughout this discussion, we emphasized the importance of understanding the relationship between the slopes of perpendicular lines. The fact that the slopes of perpendicular lines are negative reciprocals of each other is crucial for solving problems involving perpendicular bisectors. We also highlighted the usefulness of the point-slope form in finding the equation of a line when a point on the line and its slope are known. The slope-intercept form, on the other hand, provides a clear representation of the line's slope and y-intercept, which can be helpful for graphing and comparing lines. By combining these concepts and techniques, we were able to solve the given problem effectively. The equation y = -4x - 6 represents the perpendicular bisector of the line segment with a midpoint at (-1, -2). This equation can be used to further analyze the properties of the line segment and its perpendicular bisector, such as finding other points on the bisector or determining its distance from other points in the coordinate plane. Understanding the concepts and methods discussed in this guide is essential for mastering coordinate geometry and solving related problems. The ability to find the equation of a perpendicular bisector is a valuable skill in various mathematical contexts, including geometry, calculus, and linear algebra. By practicing and applying these techniques, you can enhance your problem-solving abilities and deepen your understanding of mathematical concepts.