Identifying Perpendicular Lines A Comprehensive Guide
In mathematics, particularly in coordinate geometry, understanding the relationship between lines is crucial. One fundamental concept is perpendicularity. Two lines are perpendicular if they intersect at a right angle (90 degrees). This article provides a detailed exploration of how to identify lines perpendicular to a given line, focusing on the relationship between their slopes. We will dissect the given problem, offering a step-by-step guide to determine which lines are perpendicular to the line y - 1 = (1/3)(x + 2). To achieve a comprehensive understanding, this guide will delve into the core principles of slopes, equations of lines, and the criteria for perpendicularity. The content will be structured to facilitate both learning and practical application, ensuring that you are well-equipped to tackle similar problems.
Understanding the Slope-Intercept Form
To effectively determine perpendicularity, we must first understand the slope-intercept form of a linear equation, which is y = mx + b, where m represents the slope and b represents the y-intercept. The slope, often denoted as m, quantifies the steepness and direction of a line. It is defined as the change in y divided by the change in x (rise over run). The y-intercept, denoted as b, is the point where the line crosses the y-axis. Transforming a linear equation into slope-intercept form is a crucial step in identifying its slope and, consequently, determining whether two lines are perpendicular.
The given equation, y - 1 = (1/3)(x + 2), needs to be converted into slope-intercept form to easily identify its slope. By distributing the (1/3) and adding 1 to both sides, we get: y = (1/3)x + (2/3) + 1, which simplifies to y = (1/3)x + (5/3). Therefore, the slope of the given line is 1/3. This slope is the foundation for finding lines perpendicular to the given one. The process of converting equations into slope-intercept form is a cornerstone of linear algebra, enabling straightforward comparison of slopes and y-intercepts. Mastering this conversion is essential for solving a wide range of problems involving linear equations and their graphical representations.
The Criterion for Perpendicularity
The key to identifying perpendicular lines lies in understanding their slopes. Two lines are perpendicular if and only if the product of their slopes is -1. This means if one line has a slope of m, a line perpendicular to it will have a slope of -1/m. This relationship stems from the geometric properties of right angles and the way slopes define the orientation of lines in a coordinate plane. For our given line with a slope of 1/3, a line perpendicular to it must have a slope of -1 / (1/3), which equals -3. This principle is the cornerstone of our analysis. Understanding and applying this principle allows us to quickly assess whether lines are perpendicular by simply comparing their slopes. This relationship holds true for all non-vertical lines, providing a powerful tool for analyzing geometric relationships between lines.
Now, we can evaluate each of the provided lines to see if their slopes meet this criterion. This involves converting each equation into slope-intercept form (if necessary) and then comparing its slope to -3. This step-by-step evaluation will determine which lines are indeed perpendicular to the original line. The perpendicularity criterion not only helps in identifying perpendicular lines but also in constructing lines that are perpendicular to a given line, making it a vital concept in various mathematical and real-world applications.
Analyzing the Provided Lines
Now, let's analyze each of the provided lines to determine which ones are perpendicular to the given line y - 1 = (1/3)(x + 2), which we've established has a slope of 1/3. We are looking for lines with a slope of -3.
- Line 1: y + 2 = -3(x - 4)
This equation is in point-slope form. To find the slope, we can rewrite it in slope-intercept form: y = -3x + 12 - 2, which simplifies to y = -3x + 10. The slope of this line is -3. Since -3 is the negative reciprocal of 1/3, this line is perpendicular to the given line.
- Line 2: y - 5 = 3(x + 11)
Converting this to slope-intercept form: y = 3x + 33 + 5, which simplifies to y = 3x + 38. The slope of this line is 3, not -3. Therefore, this line is not perpendicular to the given line.
- Line 3: y = -3x - 5/3
This equation is already in slope-intercept form. The slope is -3, which is the negative reciprocal of 1/3. Thus, this line is perpendicular to the given line.
- Line 4: y = (1/3)x - 2
This equation is in slope-intercept form. The slope is 1/3, which is the same as the slope of the original line. Lines with the same slope are parallel, not perpendicular. Therefore, this line is not perpendicular to the given line.
- Line 5: 3x + y = 7
To find the slope, we rewrite this equation in slope-intercept form: y = -3x + 7. The slope of this line is -3, which is the negative reciprocal of 1/3. Hence, this line is perpendicular to the given line.
This detailed analysis of each line, converting them into slope-intercept form and comparing their slopes, provides a clear methodology for determining perpendicularity. This process highlights the importance of understanding the relationship between slopes and the geometric properties of lines.
Conclusion and Key Takeaways
In conclusion, the lines perpendicular to the given line y - 1 = (1/3)(x + 2) are: y + 2 = -3(x - 4), y = -3x - 5/3, and 3x + y = 7. This determination was made by identifying the slope of the original line (1/3) and finding lines with slopes that are negative reciprocals of it (-3). The process involved converting equations into slope-intercept form to easily identify their slopes.
Key Takeaways:
- Perpendicular Lines: Two lines are perpendicular if the product of their slopes is -1.
- Slope-Intercept Form: The form y = mx + b is essential for identifying the slope (m) of a line.
- Negative Reciprocal: The slope of a line perpendicular to a line with slope m is -1/m.
- Step-by-Step Analysis: Converting equations to slope-intercept form and comparing slopes is a reliable method for determining perpendicularity.
Understanding these concepts and applying them methodically enables you to solve a wide range of problems involving perpendicular lines in coordinate geometry. The ability to identify perpendicular lines is a fundamental skill in mathematics with applications in various fields, including engineering, physics, and computer graphics. This guide has provided a comprehensive approach to identifying perpendicular lines, ensuring a solid foundation for further exploration in mathematics and related disciplines.
