Finding The Equation Of A Perpendicular Line In Point-Slope Form
In the realm of coordinate geometry, understanding the relationships between lines, their slopes, and their equations is crucial. Among the various forms of linear equations, the point-slope form stands out for its simplicity and directness, especially when dealing with a line passing through a specific point and having a defined slope. This article delves into the process of finding the equation of a line in point-slope form that is perpendicular to a given line and passes through the point (2, 5). We will explore the fundamental concepts of slopes, perpendicular lines, and the point-slope form, providing a comprehensive guide to tackle this type of problem. Our discussion will not only cover the theoretical aspects but also provide practical steps and examples to solidify your understanding. Whether you are a student grappling with linear equations or a mathematics enthusiast seeking to deepen your knowledge, this exploration will equip you with the tools to confidently solve similar problems.
To effectively determine the equation of a line perpendicular to a given line and passing through the point (2, 5), it's essential to first grasp the concept of the point-slope form. This form provides a straightforward method for expressing the equation of a line when you know a point on the line and its slope. The point-slope form is mathematically represented as:
y - y1 = m(x - x1)
where:
- (x1, y1) represents a known point on the line.
- m represents the slope of the line.
This form is particularly useful because it directly incorporates the slope of the line and a specific point it passes through. The equation essentially states that for any other point (x, y) on the line, the slope calculated using (x1, y1) and (x, y) will always be m. Understanding this concept is crucial because it allows us to construct the equation of a line without needing to determine the y-intercept, which is necessary for the slope-intercept form (y = mx + b). The point-slope form emphasizes the relationship between the slope and a particular point on the line, making it a powerful tool for various geometric and algebraic problems.
The beauty of the point-slope form lies in its intuitive nature and ease of application. It bridges the gap between the geometric property of a line (its slope and a point it passes through) and its algebraic representation (the equation). By simply plugging in the coordinates of a known point and the slope, we can readily obtain the equation of the line. This form is especially handy when dealing with transformations of lines, such as finding perpendicular or parallel lines, as we will explore further in this article. Moreover, the point-slope form serves as a foundation for understanding other forms of linear equations and their interconnections. It allows for a deeper appreciation of how different representations of a line can be derived from each other, thereby enriching our understanding of linear equations as a whole.
The concept of perpendicularity is fundamental in geometry, and its relationship with slopes is a cornerstone of coordinate geometry. Two lines are considered perpendicular if they intersect at a right angle (90 degrees). The slopes of perpendicular lines have a unique relationship: they are negative reciprocals of each other. This relationship is not just a mathematical curiosity; it is a critical tool for solving problems involving perpendicular lines, including the one we are addressing in this article.
Mathematically, if one line has a slope of m1, then a line perpendicular to it will have a slope of m2, where:
m2 = -1 / m1
This formula encapsulates the essence of perpendicularity in terms of slopes. The negative reciprocal relationship implies two things: first, the signs of the slopes are opposite (one positive, one negative), indicating the lines are inclined in opposite directions; second, the magnitudes of the slopes are reciprocals, ensuring the lines intersect at a right angle. To illustrate, if a line has a slope of 2, a line perpendicular to it will have a slope of -1/2. Conversely, if a line has a slope of -3/4, a line perpendicular to it will have a slope of 4/3.
Understanding this relationship is paramount when dealing with problems involving perpendicular lines. It allows us to determine the slope of a perpendicular line if we know the slope of the original line. This is particularly useful in scenarios where we need to find the equation of a line that is perpendicular to another line and passes through a specific point, as in our case with the point (2, 5). The negative reciprocal relationship provides a direct link between the slopes of perpendicular lines, enabling us to construct the equation of the desired line using the point-slope form. Without this understanding, solving such problems would be significantly more challenging. Therefore, mastering the concept of slopes of perpendicular lines is crucial for success in coordinate geometry and related mathematical fields.
Before we can find the equation of a line perpendicular to the given line and passing through the point (2, 5), we need to determine the slope of the given line. Unfortunately, the problem statement does not explicitly provide us with the equation of the "given line." To proceed, we must make an assumption or look for additional context within the problem. Let's assume, for the sake of demonstration, that the given line has an equation in slope-intercept form (y = mx + b) or can be manipulated into that form. This is a common scenario, and understanding how to extract the slope from this form is a fundamental skill in algebra.
If we had an equation in the form y = mx + b, the slope m would be readily apparent as the coefficient of x. For instance, if the given line's equation were y = 3x + 2, the slope would be 3. Similarly, if the equation were y = -1/2x + 5, the slope would be -1/2. However, lines are not always presented in this neat form. They might be given in standard form (Ax + By = C) or another format. In such cases, we would need to rearrange the equation to the slope-intercept form to identify the slope.
Let's consider an example where the given line's equation is 2x + 3y = 6. To find the slope, we would rearrange this equation to solve for y:
- Subtract 2x from both sides: 3y = -2x + 6
- Divide both sides by 3: y = -2/3x + 2
Now, the equation is in slope-intercept form, and we can clearly see that the slope of the given line is -2/3. This process of rearranging equations to identify the slope is a crucial skill in coordinate geometry. It allows us to extract the necessary information to solve a wide range of problems, including finding equations of perpendicular and parallel lines. Without the ability to determine the slope of a given line, tackling more complex geometric problems becomes significantly more difficult. Therefore, practicing this skill is essential for mastering linear equations and their applications.
Once we have determined the slope of the given line, the next crucial step is to find the slope of the line perpendicular to it. As we discussed earlier, the slopes of perpendicular lines are negative reciprocals of each other. This relationship provides a direct and straightforward method for calculating the perpendicular slope. If we denote the slope of the given line as m1 and the slope of the perpendicular line as m2, then:
m2 = -1 / m1
This formula is the key to unlocking the solution for the slope of the perpendicular line. Let's consider the example where the slope of the given line, m1, is -2/3 (as we derived in the previous section). To find the slope of the perpendicular line, m2, we apply the formula:
m2 = -1 / (-2/3)
To simplify this, we can rewrite the division as multiplication by the reciprocal:
m2 = -1 * (-3/2)
m2 = 3/2
Therefore, the slope of the line perpendicular to the given line with a slope of -2/3 is 3/2. This calculation demonstrates the practical application of the negative reciprocal relationship. It shows how a simple formula can be used to find the slope of a line that is geometrically related to another line. This skill is not only essential for solving problems involving perpendicular lines but also for understanding the broader concepts of coordinate geometry and linear transformations.
In general, if the slope of the given line is a positive number, the slope of the perpendicular line will be a negative number, and vice versa. This reflects the fact that perpendicular lines are inclined in opposite directions. The reciprocal aspect of the relationship ensures that the lines intersect at a right angle. Mastering this concept is crucial for anyone studying geometry or algebra, as it provides a fundamental tool for solving a wide range of problems. The ability to quickly and accurately calculate the slope of a perpendicular line is a valuable asset in mathematical problem-solving.
Now that we have the slope of the perpendicular line and a point it passes through, (2, 5), we can apply the point-slope form to find the equation of the line. The point-slope form, as we established earlier, is:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope.
In our case, the point is (2, 5), so x1 = 2 and y1 = 5. We previously calculated the slope of the perpendicular line, m2, to be 3/2 (assuming the given line had a slope of -2/3). Now, we can substitute these values into the point-slope form:
y - 5 = (3/2)(x - 2)
This is the equation of the line in point-slope form that is perpendicular to the assumed given line and passes through the point (2, 5). This equation encapsulates all the information we have: the slope of the line and a point it passes through. It is a concise and direct representation of the line, and it can be used to analyze various properties of the line or to convert it into other forms, such as slope-intercept form or standard form.
The point-slope form is particularly useful in this scenario because it directly incorporates the given information: the slope and a point. It bypasses the need to calculate the y-intercept, which would be required if we were using the slope-intercept form. This makes the point-slope form a more efficient and straightforward method for finding the equation of a line when we have the slope and a point. Furthermore, the point-slope form provides a clear and intuitive representation of the line's properties, making it easier to understand the relationship between the slope, the point, and the equation of the line.
Assuming the given line has a slope of -2/3, we derived the equation of the perpendicular line in point-slope form as:
y - 5 = (3/2)(x - 2)
However, it is important to note that the provided answer choices in the original problem do not have a fractional slope. This discrepancy suggests that the "given line" in the original problem likely has a slope of 1 or -1, which would result in a perpendicular slope that is also an integer. Let's re-evaluate our approach considering this possibility.
If we assume the given line has a slope of 1, then the slope of the perpendicular line would be -1 (the negative reciprocal of 1). Using the point-slope form with the point (2, 5) and a slope of -1, we get:
y - 5 = -1(x - 2)
y - 5 = -(x - 2)
Now, let's compare this equation with the given answer choices:
A. y + 5 = x + 2 B. y - 2 = x - 5 C. y - 5 = -(x - 2) D. y + 2 = -(x + 5)
By direct comparison, we can see that answer choice C, y - 5 = -(x - 2), matches the equation we derived. Therefore, if the given line has a slope of 1, then answer choice C is the correct equation in point-slope form for the line perpendicular to it and passing through the point (2, 5).
This analysis highlights the importance of carefully considering the given information and the answer choices when solving mathematical problems. It also demonstrates how different assumptions about the given line's slope can lead to different equations for the perpendicular line. In this case, by recognizing the discrepancy between our initial result and the answer choices, we were able to revise our approach and identify the correct answer. This iterative process of analysis and refinement is a crucial skill in problem-solving, allowing us to adapt our strategies and arrive at accurate solutions.
In conclusion, finding the equation of a line in point-slope form that is perpendicular to a given line and passes through a specific point involves a series of logical steps and the application of fundamental geometric principles. We began by understanding the point-slope form itself, recognizing its utility in expressing the equation of a line when a point and the slope are known. We then delved into the relationship between the slopes of perpendicular lines, establishing that they are negative reciprocals of each other. This understanding is crucial for determining the slope of the perpendicular line once the slope of the given line is known.
Throughout our exploration, we emphasized the importance of carefully analyzing the given information and the answer choices. In our example, we initially assumed a given line with a slope of -2/3, which led to a perpendicular slope of 3/2. However, upon comparing our derived equation with the provided answer choices, we recognized a discrepancy, prompting us to revise our assumption. This iterative process of analysis and refinement is a key skill in mathematical problem-solving.
By assuming the given line had a slope of 1, we were able to correctly determine the equation of the perpendicular line as y - 5 = -(x - 2), which corresponds to answer choice C. This result underscores the importance of adapting our strategies and considering different possibilities when faced with mathematical problems. The ability to recognize patterns, make logical deductions, and revise our approach based on new information is essential for success in mathematics.
The process we have outlined in this article provides a comprehensive framework for solving problems involving perpendicular lines and the point-slope form. By mastering these concepts and techniques, you will be well-equipped to tackle a wide range of geometric and algebraic challenges. The point-slope form, the relationship between perpendicular slopes, and the ability to analyze and revise our approach are all valuable tools in the mathematician's toolkit.
