Finding The Equation Of A Perpendicular Line Passing Through A Point
Introduction
In the realm of coordinate geometry, understanding the relationships between lines is crucial. One fundamental concept is the perpendicularity of lines. Two lines are perpendicular if they intersect at a right angle (90 degrees). This article delves into the process of finding the equation of a line that is perpendicular to a given line and passes through a specific point. We will explore the underlying principles, step-by-step methods, and practical examples to solidify your understanding. This is essential knowledge for students, educators, and anyone involved in fields that utilize geometric principles.
Understanding Perpendicular Lines
To effectively determine the equation of a perpendicular line, we must first grasp the concept of slope. The slope of a line, often denoted as m, quantifies its steepness and direction. It is defined as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. Mathematically, if we have two points (x₁, y₁) and (x₂, y₂), the slope m is calculated as:
m = (y₂ - y₁) / (x₂ - x₁)
The slope plays a pivotal role in identifying the relationship between two lines. If two lines are parallel, they have the same slope. However, when two lines are perpendicular, their slopes have a unique relationship: they are negative reciprocals of each other. This means that if one line has a slope of m, the slope of a line perpendicular to it will be -1/m. This negative reciprocal relationship is the cornerstone of finding perpendicular lines.
Consider a line with the equation y = mx + b, where m represents the slope and b represents the y-intercept (the point where the line crosses the y-axis). A line perpendicular to this line will have a slope of -1/m. For instance, if a line has a slope of 2, a perpendicular line will have a slope of -1/2. This concept is vital for solving problems involving perpendicularity.
Determining the Slope of the Perpendicular Line
Let's apply this concept to the given problem. We are given the line y = 2x + 4. This equation is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. By comparing this equation to the slope-intercept form, we can readily identify the slope of the given line as 2. Now, to find the slope of a line perpendicular to this one, we need to calculate the negative reciprocal of 2. The negative reciprocal of 2 is -1/2. Therefore, the slope of the perpendicular line is -1/2. This crucial step sets the stage for finding the equation of the perpendicular line.
Understanding how to calculate the negative reciprocal is essential. If the original slope is a fraction, say a/b, its negative reciprocal is -b/a. If the original slope is a whole number, like 2, we can consider it as 2/1, and its negative reciprocal is -1/2. This ability to manipulate slopes and find their negative reciprocals is a fundamental skill in coordinate geometry, and it is frequently used in various mathematical and real-world applications.
Using the Point-Slope Form
Now that we have the slope of the perpendicular line (-1/2) and a point it passes through (4, 6), we can use the point-slope form of a linear equation to find its equation. The point-slope form is a powerful tool for representing a line when you know its slope and a point on the line. The point-slope form is given by:
y - y₁ = m(x - x₁)
where m is the slope, and (x₁, y₁) is the given point. In our case, m = -1/2 and (x₁, y₁) = (4, 6). Plugging these values into the point-slope form, we get:
y - 6 = (-1/2)(x - 4)
This equation represents the line with a slope of -1/2 that passes through the point (4, 6). However, to express the equation in slope-intercept form (y = mx + b), we need to simplify and rearrange the equation.
Converting to Slope-Intercept Form
To convert the equation from point-slope form to slope-intercept form, we need to distribute the slope and isolate y. Starting with the equation:
y - 6 = (-1/2)(x - 4)
First, distribute the -1/2 to both terms inside the parentheses:
y - 6 = (-1/2)x + 2
Next, add 6 to both sides of the equation to isolate y:
y = (-1/2)x + 2 + 6
Finally, combine the constants:
y = (-1/2)x + 8
This is the equation of the line in slope-intercept form. We can see that the slope is -1/2, as we calculated earlier, and the y-intercept is 8. This equation represents the line that is perpendicular to y = 2x + 4 and passes through the point (4, 6). This process of converting from point-slope form to slope-intercept form is a fundamental skill in algebra and is frequently used in solving linear equations and graphing lines.
The Solution
Based on our calculations, the equation of the line that is perpendicular to y = 2x + 4 and passes through the point (4, 6) is:
y = (-1/2)x + 8
Comparing this result with the given options, we find that the correct answer is option D:
D. y = (-1/2)x + 8
This solution demonstrates the step-by-step process of finding the equation of a perpendicular line, from identifying the slope to using the point-slope form and converting to slope-intercept form. Understanding these steps is crucial for mastering coordinate geometry and solving related problems.
Conclusion
In summary, finding the equation of a line perpendicular to a given line involves several key steps. First, identify the slope of the given line. Second, calculate the negative reciprocal of that slope to find the slope of the perpendicular line. Third, use the point-slope form with the perpendicular slope and the given point. Finally, convert the equation to slope-intercept form for a clear representation of the line. This process not only provides the equation of the perpendicular line but also reinforces fundamental concepts in coordinate geometry. Mastering these concepts is essential for success in mathematics and its applications in various fields.
This process of finding perpendicular lines has applications beyond textbook problems. It is used in computer graphics, engineering, and physics to solve problems related to angles, distances, and spatial relationships. For example, architects use perpendicular lines to design buildings and structures, and engineers use them to calculate forces and stresses in mechanical systems. The ability to find and manipulate perpendicular lines is a valuable skill in many disciplines.
By understanding the relationship between slopes and the point-slope form, you can confidently tackle a wide range of problems involving lines and their properties. This knowledge forms a solid foundation for more advanced topics in mathematics and related fields. The process of solving this problem reinforces the importance of understanding fundamental concepts and applying them systematically to achieve the desired result. This approach is not only valuable in mathematics but also in problem-solving in general.
