What Is The Equation Of The Perpendicular Bisector Of A Line Segment With Midpoint (3,1) In Slope-intercept Form?
In the realm of coordinate geometry, understanding the properties of line segments and their bisectors is fundamental. This article delves into the concept of the perpendicular bisector of a line segment, focusing on a specific case where the midpoint is given as (3, 1). We will explore the steps involved in determining the equation of this perpendicular bisector in slope-intercept form. By grasping these principles, you'll be equipped to tackle various geometry problems involving lines, segments, and their relationships.
Understanding the Problem
To begin, let's clearly define the problem at hand. We are given that a line segment has a midpoint at the point (3, 1). Our goal is to find the equation of the perpendicular bisector of this line segment. Recall that a perpendicular bisector is a line that intersects the line segment at its midpoint and forms a right angle (90 degrees) with it. To achieve this, we need to determine two crucial pieces of information: the slope of the perpendicular bisector and a point that lies on it. Since the perpendicular bisector passes through the midpoint of the original line segment, we already have a point: (3, 1). The remaining challenge is to find the slope.
Finding the Slope of the Perpendicular Bisector
Finding the slope of the perpendicular bisector requires a bit more work. We'll use the fact that perpendicular lines have slopes that are negative reciprocals of each other. This means that if we can find the slope of the original line segment, we can easily determine the slope of its perpendicular bisector. Let's denote the slope of the original line segment as m. The slope of the perpendicular bisector will then be -1/m. However, we are not directly given the slope of the original line segment. We only know its midpoint. To overcome this, we need additional information about the line segment, such as another point on the segment. Let's assume, for the sake of illustration, that we have another point on the line segment, say (x₁, y₁). With two points, we can calculate the slope m using the slope formula: m = (y₂ - y₁) / (x₂ - x₁). Once we have m, we can find the slope of the perpendicular bisector as -1/m. If we don't have another point, we'll need to leave the slope of the perpendicular bisector in terms of m for now and proceed with caution, keeping in mind that our final equation will depend on this unknown slope.
Determining the Equation in Slope-Intercept Form
With the slope of the perpendicular bisector (-1/m) and a point (3, 1) on the line, we can now write the equation of the perpendicular bisector in point-slope form. The point-slope form of a line is given by y - y₁ = m( x - x₁), where (x₁, y₁) is a point on the line and m is the slope. Substituting the values we have, we get: y - 1 = (-1/m)(x - 3). To obtain the slope-intercept form (y = mx + b), we need to simplify this equation and solve for y. Distribute the slope (-1/m) on the right side: y - 1 = (-1/m)x + 3/m. Add 1 to both sides to isolate y: y = (-1/m)x + 3/m + 1. Now, the equation is in slope-intercept form. The slope is -1/m, and the y-intercept is 3/m + 1. This is the general form of the equation of the perpendicular bisector given the midpoint (3, 1) and the slope m of the original line segment.
Example and Step-by-Step Solution
Let's consider a specific example to illustrate the process. Suppose the original line segment connects the points (1, -2) and (5, 4). We are given that the midpoint is (3, 1), which we can verify using the midpoint formula: ((1 + 5)/2, (-2 + 4)/2) = (3, 1). Our goal remains the same: to find the equation of the perpendicular bisector in slope-intercept form.
Step 1: Calculate the Slope of the Original Line Segment
Using the slope formula, m = (y₂ - y₁) / (x₂ - x₁), with the points (1, -2) and (5, 4), we get: m = (4 - (-2)) / (5 - 1) = 6 / 4 = 3/2. So, the slope of the original line segment is 3/2.
Step 2: Determine the Slope of the Perpendicular Bisector
The slope of the perpendicular bisector is the negative reciprocal of the original line segment's slope. Therefore, the slope of the perpendicular bisector is -1 / (3/2) = -2/3.
Step 3: Use the Point-Slope Form
We have the slope of the perpendicular bisector (-2/3) and a point on the line, which is the midpoint (3, 1). Using the point-slope form, y - y₁ = m( x - x₁), we get: y - 1 = (-2/3)(x - 3).
Step 4: Convert to Slope-Intercept Form
Simplify the equation to slope-intercept form (y = mx + b): y - 1 = (-2/3)x + 2/3 * 3 y - 1 = (-2/3)x + 2 Add 1 to both sides: y = (-2/3)x + 2 + 1 y = (-2/3)x + 3 So, the equation of the perpendicular bisector in slope-intercept form is y = (-2/3)x + 3.
Analyzing the Answer Choices
Now, let's relate our understanding to the given answer choices. We found that the general form of the perpendicular bisector equation is y = (-1/m)x + 3/m + 1, where m is the slope of the original line segment. If we can determine the slope m from the given information or infer it from the answer choices, we can identify the correct equation. Alternatively, if we are given specific answer choices, we can analyze them based on the properties we know: the line must pass through (3, 1) and have a slope that is the negative reciprocal of the original line segment's slope.
Evaluating the Given Options
The provided options are:
A. y = (1/3) x
B. y = (1/3) x - 2
C. y = 3x
D. y = 3x - 8
To determine the correct option, we can test which of these lines passes through the midpoint (3, 1). Substitute x = 3 and y = 1 into each equation:
A. 1 = (1/3) * 3 = 1 (True)
B. 1 = (1/3) * 3 - 2 = 1 - 2 = -1 (False)
C. 1 = 3 * 3 = 9 (False)
D. 1 = 3 * 3 - 8 = 9 - 8 = 1 (True)
Options A and D both pass through the midpoint (3, 1). Now, we need to consider the slope. If the slope of the perpendicular bisector is 1/3 (as in options A and B), the slope of the original line segment would be -3. If the slope of the perpendicular bisector is 3 (as in options C and D), the slope of the original line segment would be -1/3. Without additional information about the original line segment, we cannot definitively choose between A and D. However, we can analyze further. Let's consider the slope-intercept form y = mx + b. We know that the line passes through (3, 1). For option A, y = (1/3)x: 1 = (1/3) * 3, which is true. For option D, y = 3x - 8: 1 = 3 * 3 - 8 = 1, which is also true. To differentiate, we need the slope of the original line segment. Since we don't have that information, let's work backward. If the perpendicular bisector has a slope of 1/3, the original line segment has a slope of -3. If the perpendicular bisector has a slope of 3, the original line segment has a slope of -1/3. Let's assume we have two points (x₁, y₁) and (x₂, y₂) on the original line segment. The midpoint is (3, 1). Using the midpoint formula, ( x₁ + x₂ ) / 2 = 3 and ( y₁ + y₂ ) / 2 = 1. So, x₁ + x₂ = 6 and y₁ + y₂ = 2. The slope of the original line segment is ( y₂ - y₁ ) / ( x₂ - x₁ ). If this slope is -3, then y₂ - y₁ = -3 ( x₂ - x₁ ). If this slope is -1/3, then y₂ - y₁ = (-1/3) ( x₂ - x₁ ). This analysis suggests that we need more information to definitively choose between options A and D. However, without additional context, we can proceed by assuming the simplest case. Option A, y = (1/3)x, is the simplest line that passes through the origin and has a slope of 1/3. Let's check if this is a valid perpendicular bisector.
Verifying Option A
Option A: y = (1/3)x This line has a slope of 1/3 and passes through the origin (0, 0). It also passes through (3, 1), which is the midpoint. The slope of the original line segment would be -3. Let's find a line segment with a slope of -3 and a midpoint of (3, 1). Suppose the endpoints are (x₁, y₁) and (x₂, y₂). We have:
- (x₁ + x₂) / 2 = 3
- (y₁ + y₂) / 2 = 1
- (y₂ - y₁) / ( x₂ - x₁ ) = -3
From (1) and (2), we have x₁ + x₂ = 6 and y₁ + y₂ = 2. From (3), we have y₂ - y₁ = -3 ( x₂ - x₁ ). Let's choose x₁ = 2, then x₂ = 4. So, y₂ - y₁ = -3 (4 - 2) = -6. We also have y₁ + y₂ = 2. Adding these two equations, we get 2y₂ = -4, so y₂ = -2. Then y₁ = 2 - y₂ = 2 - (-2) = 4. The endpoints are (2, 4) and (4, -2). The midpoint is ((2 + 4)/2, (4 - 2)/2) = (3, 1). The slope of the line segment is (-2 - 4) / (4 - 2) = -6 / 2 = -3. The slope of the perpendicular bisector is 1/3, which matches the slope of option A. So, y = (1/3)x is a valid perpendicular bisector.
Conclusion
In conclusion, the equation of the perpendicular bisector of the given line segment with a midpoint at (3, 1) can be determined by finding the negative reciprocal of the original line segment's slope and using the point-slope form to derive the equation in slope-intercept form. Through a step-by-step example and analysis of the answer choices, we've demonstrated the process and verified the solution. Without additional information, the simplest valid answer is option A: y = (1/3)x. This article provides a comprehensive guide to solving such problems, enhancing your understanding of coordinate geometry and the properties of lines and segments. Remember, practice is key to mastering these concepts, so try applying these techniques to various problems to solidify your knowledge.