A Given Line Segment Has A Midpoint At (-1, -2). What Is The Equation, In Slope-intercept Form, Of The Perpendicular Bisector Of This Line Segment?

Jun 19, 2025 by ADMIN 148 views

Understanding Perpendicular Bisectors and Slope-Intercept Form

In geometry, understanding perpendicular bisectors is crucial for solving various problems, especially those involving line segments and their properties. The perpendicular bisector of a line segment is a line that intersects the segment at its midpoint and forms a right angle (90 degrees) with it. This definition highlights two key characteristics: the bisecting aspect, which means it cuts the segment into two equal parts, and the perpendicular aspect, indicating the right angle formed at the intersection.

Finding the Midpoint: The journey to determining the equation of a perpendicular bisector begins with identifying the midpoint of the given line segment. The midpoint is the point that divides the segment into two equal halves. If we have the coordinates of the endpoints of the line segment, say $(x_1, y_1)$ and $(x_2, y_2)$ , we can find the midpoint using the midpoint formula:

Midpoint = $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

This formula calculates the average of the x-coordinates and the average of the y-coordinates, giving us the coordinates of the midpoint. For example, if the endpoints are $(1, 2)$ and $(5, 6)$ , the midpoint would be:

Midpoint = $\left(\frac{1 + 5}{2}, \frac{2 + 6}{2}\right) = (3, 4)$

Determining the Slope: The slope of a line is a measure of its steepness and direction. It is defined as the "rise over run," or the change in the y-coordinate divided by the change in the x-coordinate. Given two points on a line, $(x_1, y_1)$ and $(x_2, y_2)$ , the slope ( extit{m}) can be calculated using the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

For instance, if we have the points $(1, 2)$ and $(5, 6)$ , the slope of the line passing through these points would be:

$m = \frac{6 - 2}{5 - 1} = \frac{4}{4} = 1$

The concept of perpendicularity introduces an important relationship between the slopes of two lines. If two lines are perpendicular, the product of their slopes is -1. In other words, if a line has a slope of extit{m}, the slope of a line perpendicular to it will be $-\frac{1}{m}$ . This is known as the negative reciprocal. For example, if a line has a slope of 2, a line perpendicular to it will have a slope of $-\frac{1}{2}$ .

Perpendicular bisectors combine these concepts. The perpendicular bisector not only passes through the midpoint of a line segment but also is perpendicular to it. Therefore, to find the equation of a perpendicular bisector, we need to determine the midpoint of the line segment and the negative reciprocal of the slope of the original line segment.

The slope-intercept form is a fundamental way to represent a linear equation. It provides a clear and concise understanding of the line's characteristics, specifically its slope and y-intercept. The general form of the slope-intercept equation is:

$y = mx + b$

where:

  extit{y} represents the y-coordinate of any point on the line.

  extit{x} represents the x-coordinate of any point on the line.

  extit{m} represents the slope of the line, indicating its steepness and direction.

  extit{b} represents the y-intercept, the point where the line crosses the y-axis.

The slope ( extit{m}) is a critical parameter in this form. As discussed earlier, it is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. A positive slope indicates that the line rises as you move from left to right, while a negative slope indicates that the line falls. A slope of zero represents a horizontal line, and an undefined slope represents a vertical line.

The y-intercept ( extit{b}) is the point where the line intersects the y-axis. At this point, the x-coordinate is always zero. The y-intercept provides a fixed reference point for the line's position on the coordinate plane. It is the value of extit{y} when extit{x} is zero.

Converting to Slope-Intercept Form: Often, linear equations are given in other forms, such as the standard form (Ax + By = C). To effectively analyze and work with these equations, it's essential to convert them to slope-intercept form. This involves isolating extit{y} on one side of the equation. Let's illustrate this with an example:

Consider the equation: $2x + 3y = 6$

To convert this to slope-intercept form, we follow these steps:

Subtract $2x$ from both sides: $3y = -2x + 6$
Divide both sides by 3: $y = -\frac{2}{3}x + 2$

Now, the equation is in slope-intercept form, where the slope ( extit{m}) is $-\frac{2}{3}$ and the y-intercept ( extit{b}) is 2.

Using Slope-Intercept Form: The slope-intercept form is incredibly useful for various tasks, including:

Graphing Lines: Given the slope and y-intercept, you can easily plot the line on a coordinate plane. Start by plotting the y-intercept, and then use the slope to find another point on the line. For example, if the slope is $\frac{1}{2}$ , move 1 unit up and 2 units to the right from the y-intercept to find another point.
Identifying Slope and Y-Intercept: Directly reading the slope and y-intercept from the equation makes it simple to understand the line's behavior.
Writing Equations: If you know the slope and y-intercept of a line, you can immediately write its equation in slope-intercept form.
Comparing Lines: You can quickly compare the slopes and y-intercepts of different lines to determine if they are parallel, perpendicular, or intersecting.

In summary, the slope-intercept form provides a powerful tool for understanding and manipulating linear equations. Its clear representation of the slope and y-intercept makes it invaluable for graphing, analysis, and problem-solving in mathematics.

To find the equation of the perpendicular bisector of a line segment in slope-intercept form, we need to follow a series of logical steps that combine our understanding of midpoints, slopes, perpendicularity, and the slope-intercept form itself. This process ensures we accurately capture the line that both bisects the segment and is perpendicular to it.

Step 1: Find the Midpoint of the Line Segment: As the perpendicular bisector passes through the midpoint of the given line segment, our first task is to determine the coordinates of this midpoint. The midpoint formula, which we discussed earlier, is our key tool here:

Midpoint = $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the endpoints of the line segment. This formula averages the x-coordinates and the y-coordinates to locate the exact center point of the segment. The midpoint will be a crucial point that lies on the perpendicular bisector, which we'll use later in our calculations.

Step 2: Calculate the Slope of the Original Line Segment: The slope of the original line segment is essential because the perpendicular bisector is, by definition, perpendicular to it. We use the slope formula to find the slope ( extit{m}) of the original line segment:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

where, again, $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the endpoints. The slope tells us the steepness and direction of the line segment, which we'll need to determine the slope of the perpendicular bisector.

Step 3: Determine the Slope of the Perpendicular Bisector: Since the perpendicular bisector is perpendicular to the original line segment, its slope is the negative reciprocal of the original segment's slope. If the slope of the original line segment is extit{m}, then the slope of the perpendicular bisector ( extit{m_perp}) is:

$m_{perp} = -\frac{1}{m}$

This negative reciprocal relationship ensures that the two lines intersect at a right angle. The new slope, extit{m_perp}, is a critical value for constructing the equation of the perpendicular bisector.

Step 4: Use the Point-Slope Form to Create the Equation: Now that we have the slope of the perpendicular bisector ( extit{m_perp}) and a point it passes through (the midpoint we found in Step 1), we can use the point-slope form of a linear equation to create the equation of the perpendicular bisector. The point-slope form is:

$y - y_1 = m_{perp}(x - x_1)$

where $(x_1, y_1)$ is the midpoint, and extit{m_perp} is the slope of the perpendicular bisector. By substituting the known values into this form, we obtain an equation that describes the perpendicular bisector.

Step 5: Convert the Equation to Slope-Intercept Form: The final step is to convert the equation from point-slope form to slope-intercept form ( $y = mx + b$ ). This involves distributing and simplifying the equation to isolate extit{y} on one side. The resulting equation will clearly show the slope and y-intercept of the perpendicular bisector, making it easy to graph and analyze.

By following these five steps systematically, we can confidently determine the equation of the perpendicular bisector of any given line segment in slope-intercept form. Each step builds upon the previous one, combining geometric concepts and algebraic techniques to arrive at the final equation.

To solidify our understanding of how to find the equation of a perpendicular bisector in slope-intercept form, let's walk through a detailed example. This will demonstrate how the steps we outlined come together in a practical application.

Problem: Suppose we are given a line segment with endpoints A(1, 3) and B(5, 1). Our goal is to find the equation of the perpendicular bisector of this segment in slope-intercept form.

Step 1: Find the Midpoint of the Line Segment: We begin by finding the midpoint of the line segment AB. Using the midpoint formula:

Midpoint = $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Substituting the coordinates of A and B, we get:

Midpoint = $\left(\frac{1 + 5}{2}, \frac{3 + 1}{2}\right) = (3, 2)$

So, the midpoint of the line segment AB is (3, 2). This point lies on the perpendicular bisector.

Step 2: Calculate the Slope of the Original Line Segment: Next, we need to find the slope of the original line segment AB. Using the slope formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the coordinates of A and B, we get:

$m = \frac{1 - 3}{5 - 1} = \frac{-2}{4} = -\frac{1}{2}$

The slope of the line segment AB is $-\frac{1}{2}$ .

Step 3: Determine the Slope of the Perpendicular Bisector: The perpendicular bisector has a slope that is the negative reciprocal of the original segment's slope. Therefore, the slope of the perpendicular bisector ( extit{m_perp}) is:

$m_{perp} = -\frac{1}{m} = -\frac{1}{(-\frac{1}{2})} = 2$

The slope of the perpendicular bisector is 2.

Step 4: Use the Point-Slope Form to Create the Equation: Now we use the point-slope form of a linear equation with the midpoint (3, 2) and the slope of the perpendicular bisector (2):

$y - y_1 = m_{perp}(x - x_1)$

Substituting the values, we get:

$y - 2 = 2(x - 3)$

This is the equation of the perpendicular bisector in point-slope form.

Step 5: Convert the Equation to Slope-Intercept Form: Finally, we convert the equation to slope-intercept form ( $y = mx + b$ ) by distributing and simplifying:

$y - 2 = 2x - 6$

Add 2 to both sides:

$y = 2x - 4$

Thus, the equation of the perpendicular bisector of the line segment AB in slope-intercept form is $y = 2x - 4$ .

This example illustrates how we combine the concepts of midpoints, slopes, and perpendicularity to find the equation of a perpendicular bisector. By following these steps, you can solve similar problems effectively.

Let's apply the concepts and steps we've discussed to solve the specific problem presented. The problem states that a given line segment has a midpoint at (-1, -2). To find the equation of the perpendicular bisector in slope-intercept form, we need additional information, specifically the slope of the original line segment. Without this, we cannot determine the slope of the perpendicular bisector.

Understanding the Challenge: We have the midpoint, which is a crucial point on the perpendicular bisector. However, to define a line, we need either another point or the slope. In this case, we need the slope. The problem does not directly provide the slope of the original line segment, nor does it give us another point on the segment. This means we must infer or derive this information from the problem statement or any accompanying context (which is not provided here).

Missing Information: To proceed, we need either:

The coordinates of the endpoints of the line segment. This would allow us to calculate the slope using the slope formula.
The equation of the original line segment in any form. From this, we could determine the slope.

Hypothetical Scenario: Let's assume, for the sake of demonstration, that the original problem included the equation of the line segment. Suppose the equation of the original line segment is $y = \frac{1}{4}x + 3$ . Now we have the necessary information to solve the problem.

Step 1: Identify the Slope of the Original Line Segment: From the equation $y = \frac{1}{4}x + 3$ , we can see that the slope ( extit{m}) of the original line segment is $\frac{1}{4}$ .

Step 2: Determine the Slope of the Perpendicular Bisector: The slope of the perpendicular bisector ( extit{m_perp}) is the negative reciprocal of the original slope:

$m_{perp} = -\frac{1}{m} = -\frac{1}{\frac{1}{4}} = -4$

So, the slope of the perpendicular bisector is -4.

Step 3: Use the Point-Slope Form: We know the midpoint is (-1, -2), and the slope of the perpendicular bisector is -4. Using the point-slope form:

$y - y_1 = m_{perp}(x - x_1)$

Substituting the values, we get:

$y - (-2) = -4(x - (-1))$

$y + 2 = -4(x + 1)$

Step 4: Convert to Slope-Intercept Form: Now we convert the equation to slope-intercept form:

$y + 2 = -4x - 4$

Subtract 2 from both sides:

$y = -4x - 6$

Therefore, if the original line segment had a slope of $\frac{1}{4}$ , the equation of the perpendicular bisector would be $y = -4x - 6$ .

Matching with the Given Options: Comparing this result with the options provided:

A. $y = -4x - 4$ B. $y = -4x - 6$ C. $y = \frac{1}{4}x - \frac{9}{4}$

Option B, $y = -4x - 6$ , matches our calculated equation.

Conclusion: In the hypothetical scenario where the original line segment has a slope of $\frac{1}{4}$ , the equation of the perpendicular bisector in slope-intercept form is $y = -4x - 6$ . However, without the slope or additional information, the problem cannot be definitively solved. This example highlights the importance of having all necessary information when tackling mathematical problems.