Finding The Equation Of A Perpendicular Bisector A Step-by-Step Guide
In the realm of coordinate geometry, determining the equation of a perpendicular bisector is a fundamental skill. A perpendicular bisector, as the name suggests, is a line that intersects a given line segment at its midpoint and forms a right angle (90 degrees) with it. This article delves into the step-by-step process of finding the equation of a perpendicular bisector, using the specific example of a line segment with endpoints (-5, 3) and (5, 5). We will explore the underlying concepts, formulas, and calculations involved, providing a comprehensive understanding of this important geometric concept.
Understanding the Perpendicular Bisector
Before diving into the calculations, let's solidify our understanding of the perpendicular bisector. Imagine a line segment drawn between two points on a coordinate plane. The perpendicular bisector is a line that cuts this segment into two equal halves (bisects) and meets the segment at a perfect right angle (perpendicular). This means it has two key properties:
- It passes through the midpoint of the original line segment.
- It is perpendicular to the original line segment.
To find the equation of the perpendicular bisector, we need to leverage these two properties. We'll first determine the midpoint of the segment and then find the slope of the perpendicular bisector. With these two pieces of information, we can use the point-slope form of a linear equation to arrive at our final answer. In this article, we are going to clarify each step, and we will use the example in the prompt in the calculation process.
Step 1: Finding the Midpoint
The midpoint of a line segment is the point that lies exactly halfway between its two endpoints. If we have two points, say (x1, y1) and (x2, y2), the midpoint formula is given by:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
This formula essentially calculates the average of the x-coordinates and the average of the y-coordinates to find the middle point.Let's apply this to our problem. We have the endpoints (-5, 3) and (5, 5). Plugging these values into the midpoint formula, we get:
Midpoint = ((-5 + 5)/2, (3 + 5)/2) = (0/2, 8/2) = (0, 4)
Therefore, the midpoint of the line segment with endpoints (-5, 3) and (5, 5) is (0, 4). This point will lie on the perpendicular bisector, which is a crucial piece of information for later steps.Understanding the midpoint is essential because it serves as the anchor point for the perpendicular bisector. The perpendicular bisector must pass through this point, and knowing its coordinates allows us to use the point-slope form of a linear equation to determine the line's equation. Without the midpoint, we would lack a crucial reference point, making it impossible to define the specific location of the perpendicular bisector.In essence, the midpoint calculation provides the exact center of the original line segment, which is a fundamental component in constructing the perpendicular bisector. The accuracy of this calculation is paramount, as any error in the midpoint will propagate through subsequent steps, leading to an incorrect equation for the perpendicular bisector. This careful determination of the midpoint underscores the importance of precision in geometric problem-solving.
Step 2: Finding the Slope of the Original Line Segment
The slope of a line is a measure of its steepness and direction. It tells us how much the line rises or falls for every unit of horizontal change. The slope (m) between two points (x1, y1) and (x2, y2) is calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Let's calculate the slope of the line segment with endpoints (-5, 3) and (5, 5):
m = (5 - 3) / (5 - (-5)) = 2 / 10 = 1/5
So, the slope of the original line segment is 1/5. This value represents the inclination of the line segment in the coordinate plane. The slope calculation is a crucial step because it provides the foundation for determining the slope of the perpendicular bisector. As we know, perpendicular lines have slopes that are negative reciprocals of each other. Therefore, once we have the slope of the original line segment, we can easily find the slope of the perpendicular bisector by taking its negative reciprocal. Without accurately calculating the slope of the original line segment, we cannot proceed to find the slope of the perpendicular bisector, which is essential for determining its equation. The slope is a fundamental property of a line, and its precise determination is paramount for understanding and manipulating lines in coordinate geometry. The slope provides a numerical measure of the line's inclination, allowing us to compare and contrast different lines and to perform geometric calculations with accuracy and confidence. In the context of the perpendicular bisector, the slope is the key that unlocks the relationship between the original line segment and the line that bisects it at a right angle.
Step 3: Finding the Slope of the Perpendicular Bisector
The key property of perpendicular lines is that their slopes are negative reciprocals of each other. This means if one line has a slope of m, a line perpendicular to it will have a slope of -1/m. We've already found that the slope of the original line segment is 1/5. To find the slope of the perpendicular bisector, we take the negative reciprocal of 1/5:
Slope of perpendicular bisector = -1 / (1/5) = -5
Therefore, the slope of the perpendicular bisector is -5. This negative reciprocal relationship is a fundamental concept in coordinate geometry and is crucial for understanding the behavior of perpendicular lines. The negative reciprocal relationship between the slopes of perpendicular lines stems from the fact that the product of their slopes is always -1. This mathematical property arises from the geometric condition that perpendicular lines intersect at a right angle (90 degrees). When two lines are perpendicular, their slopes have opposite signs (one positive, one negative) and their magnitudes are reciprocals of each other. This ensures that the lines meet at a right angle, forming a perfect L-shape.Understanding this relationship allows us to easily find the slope of a line perpendicular to a given line, which is essential in various geometric constructions and problem-solving scenarios. The concept of negative reciprocals is not limited to perpendicular bisectors; it extends to any situation where perpendicular lines are involved, making it a cornerstone of coordinate geometry.
Step 4: Finding the Equation of the Perpendicular Bisector
Now that we have the slope of the perpendicular bisector (-5) and a point it passes through (the midpoint (0, 4)), we can use the point-slope form of a linear equation to find its equation. The point-slope form is:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) is a point on the line. Plugging in our values, we get:
y - 4 = -5(x - 0)
Simplifying this equation, we get:
y - 4 = -5x
Adding 4 to both sides, we get the equation of the perpendicular bisector in slope-intercept form (y = mx + b):
y = -5x + 4
Therefore, the equation of the perpendicular bisector of the line segment with endpoints (-5, 3) and (5, 5) is y = -5x + 4. This equation represents a straight line that perfectly bisects the original line segment at a right angle. The slope-intercept form of the equation (y = mx + b) provides a clear representation of the line's properties. The coefficient 'm' (-5 in this case) represents the slope, indicating the line's steepness and direction. The constant 'b' (4 in this case) represents the y-intercept, which is the point where the line crosses the y-axis. By knowing the slope and y-intercept, we can easily visualize and graph the line on the coordinate plane. The equation y = -5x + 4 encapsulates all the essential information about the perpendicular bisector, allowing us to analyze its behavior and its relationship to other geometric elements in the coordinate plane.
Final Answer
Therefore, the correct answer is:
B. y = -5x + 4
This comprehensive solution demonstrates the step-by-step process of finding the equation of a perpendicular bisector. By understanding the underlying concepts, applying the appropriate formulas, and performing the calculations with precision, we can confidently solve similar problems in coordinate geometry. The perpendicular bisector is a fundamental geometric concept with wide-ranging applications in mathematics, engineering, and computer graphics. Its ability to divide a line segment into two equal parts at a right angle makes it a crucial tool in various geometric constructions and problem-solving scenarios.
In summary, finding the equation of the perpendicular bisector involves several key steps: determining the midpoint of the original line segment, calculating the slope of the original segment, finding the negative reciprocal of that slope to obtain the slope of the perpendicular bisector, and finally, using the point-slope form to derive the equation of the perpendicular bisector. This process highlights the interconnectedness of various geometric concepts and the importance of mastering these fundamental skills for advanced mathematical studies. The perpendicular bisector serves as a building block for more complex geometric constructions and analyses, emphasizing its significance in the broader mathematical landscape.
In conclusion, the equation of the perpendicular bisector, y = -5x + 4, represents a line that not only bisects the original line segment but also does so at a perfect right angle. This geometric property makes it a valuable tool in various mathematical and practical applications, showcasing the power and elegance of coordinate geometry.
