Finding Slope Of Perpendicular Line Comprehensive Guide
In the realm of coordinate geometry, the relationship between lines, especially perpendicular lines, holds significant importance. When two lines intersect at a right angle (90 degrees), they are considered perpendicular. This perpendicularity has a direct connection to the slopes of the lines involved. Slope, a fundamental concept in linear equations, defines the steepness and direction of a line. Understanding the relationship between slopes of perpendicular lines is crucial for solving various geometric problems. In this article, we will delve into the process of determining the slope of a line perpendicular to a given line, providing a comprehensive explanation to enhance your understanding of this concept.
The cornerstone of solving this problem lies in the relationship between the slopes of perpendicular lines. The slopes of two perpendicular lines 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 inverse relationship is the key to unlocking the solution. To effectively apply this principle, we must first determine the slope of the given line. Let's explore the steps involved in finding the slope of the given line and subsequently the slope of the perpendicular line.
Our journey begins with identifying the slope of line 'p', which is defined by the equation 2y + 18x = 9. To find the slope, we need to transform this equation into slope-intercept form, which is expressed as y = mx + b. In this form, 'm' represents the slope, and 'b' represents the y-intercept. The slope-intercept form provides a clear and concise way to visualize the characteristics of a line. By rearranging the given equation, we can easily extract the slope. Let's proceed with the transformation.
To isolate 'y', we first subtract 18x from both sides of the equation, resulting in 2y = -18x + 9. Next, we divide both sides by 2 to obtain y = -9x + 4.5. Now, the equation is in slope-intercept form. By comparing this equation with the general form y = mx + b, we can clearly identify that the slope of line 'p' is -9. This slope indicates that for every unit increase in 'x', the value of 'y' decreases by 9 units. Having determined the slope of line 'p', we are now ready to calculate the slope of the line perpendicular to it.
Now that we have the slope of line 'p', which is -9, we can use the principle of negative reciprocals to find the slope of line 'r', which is perpendicular to line 'p'. The negative reciprocal of a number is obtained by flipping the fraction (taking the reciprocal) and changing its sign. In this case, the slope of line 'p' is -9, which can be expressed as a fraction -9/1. To find the negative reciprocal, we flip the fraction to get -1/9 and then change the sign, resulting in 1/9. Therefore, the slope of line 'r', which is perpendicular to line 'p', is 1/9. This positive slope indicates that for every unit increase in 'x', the value of 'y' increases by 1/9 units. This contrasting direction compared to line 'p' is a characteristic of perpendicular lines.
The final step is to verify your answer. You can verify your answer by multiplying the slopes of two perpendicular lines, and the result is -1. If the answer is correct, this will give you peace of mind.
In summary, determining the slope of a line perpendicular to a given line involves understanding the fundamental relationship between their slopes – they are negative reciprocals of each other. By converting the equation of the given line into slope-intercept form, we can easily identify its slope. Then, by taking the negative reciprocal of this slope, we can find the slope of the perpendicular line. This process demonstrates the interconnectedness of geometric concepts and the power of algebraic manipulation in solving problems. Mastering this concept is crucial for tackling more complex problems in coordinate geometry and related fields. This knowledge equips you with a valuable tool for analyzing and understanding the relationships between lines in the xy-plane.
Therefore, the slope of line r is 1/9, which corresponds to option C.
The concept of slope is fundamental in understanding lines within the realm of mathematics. It provides a measure of a line's steepness and direction on a coordinate plane. In essence, the slope tells us how much the line rises or falls for every unit of horizontal change. This ratio of vertical change to horizontal change is crucial for analyzing linear relationships and solving geometric problems. Lines with positive slopes ascend from left to right, while lines with negative slopes descend. A horizontal line has a slope of zero, indicating no vertical change, and a vertical line has an undefined slope, as its horizontal change is zero. Mastering the concept of slope is essential for comprehending the behavior and interactions of lines in various mathematical contexts. This article will delve into the intricacies of slope and its applications, particularly in the context of perpendicular lines.
Perpendicular lines hold a special place in geometry, characterized by their intersection at a right angle (90 degrees). This orthogonal relationship has significant implications for their slopes. The defining characteristic of perpendicular lines is the negative reciprocal relationship between their slopes. If one line has a slope of 'm', a line perpendicular to it will have a slope of '-1/m'. This inverse relationship is not merely a coincidence but a fundamental property of perpendicularity. Understanding this relationship allows us to determine whether two lines are perpendicular simply by examining their slopes. This concept is widely used in various fields, including architecture, engineering, and computer graphics, where precise angular relationships are crucial. Let's explore further how this negative reciprocal relationship plays out in determining the slopes of perpendicular lines.
The negative reciprocal relationship between the slopes of perpendicular lines is the cornerstone of understanding their geometric interaction. Imagine a line with a positive slope. A line perpendicular to it must descend in the opposite direction, hence the negative sign in the slope. The reciprocal part of the relationship ensures that the lines intersect at a right angle. Mathematically, this relationship can be expressed as m1 * m2 = -1, where m1 and m2 are the slopes of the two perpendicular lines. This equation provides a powerful tool for verifying perpendicularity and calculating unknown slopes. For instance, if we know the slope of one line, we can easily determine the slope of any line perpendicular to it by taking the negative reciprocal. This principle is not just a mathematical curiosity but a fundamental aspect of how lines interact in space. In the following sections, we will see how to apply this relationship to solve practical problems.
Finding the slope of a line perpendicular to a given line involves a systematic approach that leverages the negative reciprocal relationship. The first step is to determine the slope of the given line. This may involve converting the equation of the line into slope-intercept form (y = mx + b), where 'm' represents the slope. Alternatively, if we are given two points on the line, we can calculate the slope using the formula m = (y2 - y1) / (x2 - x1). Once we have the slope of the given line, the next step is to find its negative reciprocal. This is done by flipping the fraction (taking the reciprocal) and changing the sign. For example, if the slope of the given line is 2/3, the slope of the perpendicular line would be -3/2. This simple yet powerful process allows us to quickly determine the slope of any line perpendicular to a given line. Let's illustrate this process with a practical example.
Consider a line defined by the equation 3x + 4y = 12. Our goal is to find the slope of a line perpendicular to this given line. First, we need to convert the equation into slope-intercept form (y = mx + b). Subtracting 3x from both sides gives us 4y = -3x + 12. Then, dividing both sides by 4 yields y = (-3/4)x + 3. From this equation, we can see that the slope of the given line is -3/4. To find the slope of a line perpendicular to this line, we take the negative reciprocal of -3/4. This involves flipping the fraction to get -4/3 and then changing the sign, resulting in 4/3. Therefore, the slope of a line perpendicular to the given line is 4/3. This example demonstrates the step-by-step process of applying the negative reciprocal relationship to find the slope of a perpendicular line. Such practical applications highlight the importance of understanding this concept in solving geometric problems.
The knowledge of slopes and perpendicular lines extends far beyond the classroom, finding practical applications in various real-world scenarios. In architecture and engineering, understanding perpendicularity is crucial for designing stable structures and ensuring precise alignments. The walls of a building, for example, need to be perpendicular to the ground to maintain structural integrity. In computer graphics, perpendicular lines are fundamental for creating realistic 3D models and rendering images. The concept is also used in navigation and mapping, where perpendicular lines can represent directions and boundaries. Furthermore, in physics, the concept of perpendicularity is used to analyze forces and motion. The normal force, for instance, is always perpendicular to the surface of contact. These diverse applications highlight the importance of understanding slopes and perpendicular lines in various professional fields. By mastering these concepts, individuals can gain a deeper appreciation for the mathematical principles that govern the world around us.
In conclusion, understanding the relationship between slopes and perpendicular lines is a fundamental concept in mathematics with wide-ranging applications. The negative reciprocal relationship between their slopes provides a powerful tool for solving geometric problems and analyzing real-world scenarios. By mastering the steps involved in finding the slope of a perpendicular line, individuals can enhance their problem-solving skills and gain a deeper appreciation for the interconnectedness of mathematical concepts. This knowledge is not just theoretical but has practical implications in various fields, from architecture and engineering to computer graphics and physics. By understanding these principles, we can better appreciate the mathematical foundations that underpin our world.
