Finding The Perpendicular Bisector Equation Of Segment AB A(12,-3) And B(6,-15)

Finding the equation of a perpendicular bisector is a common problem in coordinate geometry. It combines concepts of midpoints, slopes, and linear equations. In this article, we will walk through the process step-by-step, ensuring a clear understanding of how to tackle such problems. Specifically, we will find the general form equation for the perpendicular bisector of the segment AB, where the coordinates of points A and B are given as A(12, -3) and B(6, -15). This process involves several key stages, each of which contributes to the final solution. First, we need to find the midpoint of the line segment AB, which will be a point on the perpendicular bisector. Then, we calculate the slope of AB, which allows us to determine the slope of the perpendicular bisector. Finally, using the point-slope form of a line equation, we can derive the equation of the perpendicular bisector and convert it into the general form. Each of these steps is crucial and interconnected, providing a systematic approach to solving geometric problems in the coordinate plane.

Step 1: Finding the Midpoint of AB

To begin, let’s find the midpoint of the line segment AB. The midpoint, denoted as M, is the point that lies exactly halfway between points A and B. The coordinates of the midpoint can be calculated using the midpoint formula, which is derived from averaging the x-coordinates and the y-coordinates of the endpoints. Given two points, A(x1, y1) and B(x2, y2), the midpoint M(xm, ym) is calculated as follows:

xm = (x1 + x2) / 2 ym = (y1 + y2) / 2

In our case, the coordinates of A are (12, -3) and the coordinates of B are (6, -15). Plugging these values into the midpoint formula, we get:

xm = (12 + 6) / 2 = 18 / 2 = 9 ym = (-3 + -15) / 2 = -18 / 2 = -9

Thus, the midpoint M of the line segment AB has coordinates (9, -9). This point is crucial because the perpendicular bisector passes through it. Understanding how to calculate the midpoint is fundamental in coordinate geometry, as it forms the basis for many geometric constructions and proofs. The midpoint not only divides the segment into two equal parts but also provides a central reference point for further analysis. This calculation is accurate and directly applies the midpoint formula, ensuring we have the correct coordinates for the next steps. This midpoint will be an essential component in determining the equation of the perpendicular bisector, as it provides a specific point that lies on the bisector, allowing us to use point-slope form in the later stages.

Step 2: Determining the Slope of AB

Next, we need to determine the slope of the line segment AB. The slope is a measure of the steepness and direction of a line, and it is defined as the change in y divided by the change in x between two points on the line. The slope, often denoted as m, can be calculated using the formula:

m = (y2 - y1) / (x2 - x1)

Given the points A(12, -3) and B(6, -15), we can plug these coordinates into the slope formula:

m_AB = (-15 - (-3)) / (6 - 12) m_AB = (-15 + 3) / (-6) m_AB = -12 / -6 m_AB = 2

Therefore, the slope of the line segment AB is 2. This slope represents how much the line rises (or falls) for each unit of horizontal change. A positive slope indicates that the line rises from left to right, while a negative slope indicates that the line falls from left to right. The slope of AB is a critical piece of information because the slope of the perpendicular bisector will be the negative reciprocal of this value. Understanding how to calculate slopes is crucial in various mathematical contexts, including linear equations, calculus, and physics. The slope provides a clear indication of a line's direction and steepness, making it an essential parameter in geometric analysis. This calculation ensures that we have the correct slope for AB, which is crucial for determining the slope of the perpendicular bisector in the subsequent steps.

Step 3: Finding the Slope of the Perpendicular Bisector

The perpendicular bisector is a line that intersects the line segment AB at a right angle (90 degrees) and passes through its midpoint. One of the key properties of perpendicular lines is that their slopes are negative reciprocals of each other. This means if the slope of line AB is m, then the slope of the perpendicular bisector, denoted as m_perp, is -1/m. We’ve already calculated the slope of AB as 2. To find the slope of the perpendicular bisector, we take the negative reciprocal:

m_perp = -1 / m_AB m_perp = -1 / 2

So, the slope of the perpendicular bisector is -1/2. This negative reciprocal relationship is a fundamental concept in geometry and is crucial for understanding the orientation of perpendicular lines. The negative reciprocal ensures that the two lines intersect at a right angle, a defining characteristic of perpendicularity. This slope will be used in conjunction with the midpoint we found earlier to determine the equation of the perpendicular bisector. The calculation of the perpendicular slope is accurate and crucial, as it directly impacts the correctness of the final equation of the perpendicular bisector. Understanding the relationship between slopes of perpendicular lines is essential in various applications, including coordinate geometry, engineering, and computer graphics. With the correct slope of the perpendicular bisector, we are now well-equipped to derive its equation.

Step 4: Deriving the Equation of the Perpendicular Bisector

Now that we have the slope of the perpendicular bisector (-1/2) and a point it passes through (the midpoint M(9, -9)), we can determine its equation. The point-slope form of a linear equation is particularly useful in this situation. The point-slope form is given by:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope of the line. Plugging in the midpoint M(9, -9) and the slope of the perpendicular bisector (-1/2), we get:

y - (-9) = (-1/2)(x - 9) y + 9 = (-1/2)(x - 9)

This equation represents the perpendicular bisector in point-slope form. However, the question asks for the general form of the equation. The general form of a linear equation is given by:

Ax + By + C = 0

where A, B, and C are constants. To convert our point-slope equation to general form, we need to eliminate the fraction and rearrange the terms:

Multiply both sides by 2 to eliminate the fraction:

2(y + 9) = 2((-1/2)(x - 9)) 2y + 18 = -(x - 9) 2y + 18 = -x + 9

Now, rearrange the equation to the general form:

x + 2y + 18 - 9 = 0 x + 2y + 9 = 0

Thus, the general form of the equation for the perpendicular bisector of the segment AB is x + 2y + 9 = 0. This equation represents a line that is both perpendicular to the original segment AB and bisects it, meaning it passes through the midpoint. The general form is a standard way to express linear equations, and it is particularly useful for comparing different lines and identifying their properties. This comprehensive derivation ensures that we have accurately converted the point-slope form to the general form, providing a clear and concise final answer. Understanding how to derive and convert between different forms of linear equations is a fundamental skill in algebra and geometry.

Conclusion

In conclusion, we have successfully found the general form of the equation for the perpendicular bisector of the segment AB, where A(12, -3) and B(6, -15). The process involved several key steps: finding the midpoint of AB, determining the slope of AB, calculating the slope of the perpendicular bisector, and finally, deriving the equation of the perpendicular bisector in general form. Each step is crucial and builds upon the previous one, demonstrating a systematic approach to solving geometric problems in the coordinate plane. The final equation, x + 2y + 9 = 0, represents the line that is both perpendicular to AB and bisects it, fulfilling the conditions for a perpendicular bisector. This problem illustrates the interconnectedness of various concepts in coordinate geometry, such as midpoints, slopes, and linear equations. Mastering these concepts is essential for further studies in mathematics and related fields. Understanding how to derive equations for geometric figures allows for precise analysis and problem-solving in various contexts. The methodology presented here can be applied to similar problems, making it a valuable tool for students and professionals alike. The systematic approach, from finding the midpoint to determining the slope and deriving the equation, ensures a clear and accurate solution. This comprehensive walkthrough highlights the importance of understanding fundamental geometric principles and their applications in coordinate geometry. This detailed explanation provides a solid foundation for tackling more complex problems in the future, reinforcing the understanding of perpendicular bisectors and their equations.