Perpendicular Lines: Find Equations Matching Y - 1 = (1/3)(x + 2)

Have you ever wondered how to identify lines that are perfectly perpendicular to each other? It's a fundamental concept in geometry, and understanding it can unlock a deeper appreciation for the relationships between lines and angles. In this article, we'll explore how to determine which lines are perpendicular to a given line, specifically focusing on the line y - 1 = (1/3)(x + 2). We'll break down the process step by step, ensuring you grasp the underlying principles and can confidently tackle similar problems.

Understanding Perpendicular Lines

Before we dive into the specifics of our example, let's establish a solid understanding of what it means for lines to be perpendicular. Perpendicular lines are lines that intersect at a right angle, which is an angle of 90 degrees. This geometric relationship has a crucial algebraic implication: the slopes of perpendicular lines are negative reciprocals of each other. In simpler terms, if one line has a slope of m, a line perpendicular to it will have a slope of -1/m. This inverse relationship is the key to identifying perpendicular lines.

To truly grasp this concept, let's delve a little deeper into the why behind it. The slope of a line, often denoted as m, represents the line's steepness or inclination. It's calculated as the change in y divided by the change in x (rise over run). When two lines are perpendicular, their slopes interact in a way that creates that perfect 90-degree angle. The negative reciprocal relationship ensures that the lines intersect at this precise angle. Imagine one line climbing gently uphill (positive slope) and the other descending steeply (negative slope); the combination creates the perpendicular intersection.

Understanding this relationship between slopes is not just a mathematical exercise; it has practical applications in various fields. Architects and engineers rely on the principles of perpendicularity to design structures with stable and balanced angles. Navigators use perpendicular lines and angles to chart courses and determine precise locations. Even in everyday life, we encounter perpendicular lines in the corners of rooms, the intersections of streets, and countless other scenarios. So, grasping this concept opens doors to a deeper understanding of the world around us.

Determining the Slope of the Given Line

Our starting point is the line given by the equation y - 1 = (1/3)(x + 2). To determine which lines are perpendicular to this line, we first need to identify its slope. The equation is currently in point-slope form, which is a useful form for identifying both the slope and a point on the line. However, to make the slope even more apparent, we can convert the equation to slope-intercept form, which is y = mx + b, where m represents the slope and b represents the y-intercept.

Let's convert the given equation to slope-intercept form. Starting with y - 1 = (1/3)(x + 2), we distribute the (1/3) on the right side: y - 1 = (1/3)x + 2/3. Next, we add 1 to both sides to isolate y: y = (1/3)x + 2/3 + 1. Combining the constants, we get y = (1/3)x + 5/3. Now, the equation is in slope-intercept form, and we can clearly see that the slope of the given line is 1/3.

Understanding the significance of the slope is crucial. It tells us the rate at which the line rises or falls for every unit increase in x. In this case, a slope of 1/3 means that for every 3 units we move to the right along the x-axis, the line goes up 1 unit along the y-axis. This gentle upward slant is a visual representation of the line's slope. Now that we know the slope of our given line, we can use the concept of negative reciprocals to find the slopes of lines that are perpendicular to it.

Finding the Perpendicular Slope

Now that we know the slope of our given line is 1/3, we can determine the slope of any line perpendicular to it. Remember, the slopes of perpendicular lines are negative reciprocals of each other. This means we need to flip the fraction (take the reciprocal) and change the sign.

The reciprocal of 1/3 is 3/1, which simplifies to 3. Changing the sign, we get -3. Therefore, the slope of any line perpendicular to y - 1 = (1/3)(x + 2) must be -3. This negative reciprocal relationship is the cornerstone of our quest to identify perpendicular lines.

Think of it this way: the original line has a gentle positive slope, while the perpendicular line needs to have a steep negative slope to intersect it at a right angle. The negative reciprocal ensures this precise angular relationship. A slope of -3 signifies that for every 1 unit we move to the right along the x-axis, the line goes down 3 units along the y-axis. This steep downward slant is the visual manifestation of a slope that is perpendicular to our original line. With this crucial piece of information in hand, we can now examine the given options and see which ones have a slope of -3.

Analyzing the Options

Now, let's examine the given options and determine which lines have a slope of -3, making them perpendicular to our original line. We'll analyze each option individually, paying close attention to its equation form and how it reveals the slope.

• A. y + 2 = -3(x - 4)

  This equation is in point-slope form, which is y - y1 = m(x - x1), where m is the slope. In this case, we can directly see that the slope is -3. Therefore, this line is perpendicular to the given line.

• B. y - 5 = 3(x + 11)

  This equation is also in point-slope form. Here, the slope is 3. Since 3 is not the negative reciprocal of 1/3, this line is not perpendicular to the given line.

• C. y = -3x - 5/3

  This equation is in slope-intercept form, y = mx + b, where m is the slope. The slope in this equation is -3. Thus, this line is perpendicular to the given line.

• D. y = (1/3)x - 2

  This equation is in slope-intercept form. The slope is 1/3. This is the same slope as the original line, meaning this line is parallel, not perpendicular.

• E. 3x + y = 7

  To determine the slope, we need to rewrite this equation in slope-intercept form. Subtracting 3x from both sides, we get y = -3x + 7. Now we can see that the slope is -3, making this line perpendicular to the given line.

Conclusion

In conclusion, the lines perpendicular to y - 1 = (1/3)(x + 2) are:

• A. y + 2 = -3(x - 4)
• C. y = -3x - 5/3
• E. 3x + y = 7

We successfully identified these lines by understanding the relationship between the slopes of perpendicular lines and applying this knowledge to analyze the given options. Remember, the key takeaway is that perpendicular lines have slopes that are negative reciprocals of each other. By mastering this concept, you'll be well-equipped to tackle a wide range of geometry problems and appreciate the elegant connections between lines, angles, and equations. For further exploration of linear equations and their properties, consider visiting resources like Khan Academy's Linear Equations section.