Finding Ordered Pairs On A Perpendicular Line

In mathematics, understanding the relationship between lines, particularly perpendicular lines, is a fundamental concept in coordinate geometry. This article delves into how to identify ordered pairs that lie on a line perpendicular to a given line. Specifically, we'll address the question: A line has a slope of $-\frac{4}{5}$. Which ordered pairs could be points on a line that is perpendicular to this line? We will explore the underlying principles, step-by-step solutions, and practical methods to tackle such problems effectively. Understanding the concept of slopes, perpendicular lines, and how to use ordered pairs to determine the properties of lines is crucial in various mathematical applications.

Understanding Slopes and Perpendicular Lines

The slope of a line is a measure of 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, the slope (${m}$) is given by the formula:

${ m = \frac{y_2 - y_1}{x_2 - x_1} }$

where (${(x_1, y_1)}$) and (${(x_2, y_2)}$) are two points on the line.

Perpendicular lines are lines that intersect at a right angle (90 degrees). A critical property of perpendicular lines is that their slopes are negative reciprocals of each other. If a line has a slope of (${m_1}$), a line perpendicular to it will have a slope of (${m_2}$) such that:

${ m_1 \times m_2 = -1 }$

This relationship is the cornerstone for determining whether two lines are perpendicular. For instance, if a line has a slope of $-\frac{4}{5}$, a line perpendicular to it will have a slope that is the negative reciprocal of $-\frac{4}{5}$. To find this, we first take the reciprocal, which is $-\frac{5}{4}$, and then change the sign to get the negative reciprocal, which is $\frac{5}{4}$. Thus, any line with a slope of $\frac{5}{4}$ is perpendicular to the given line.

The concept of slope is vital not only in mathematics but also in various real-world applications. For example, in construction and engineering, the slope is a critical factor in designing roads, bridges, and buildings. A steeper slope might indicate a more challenging path for vehicles or pedestrians, while a gentler slope can ensure ease of movement. Similarly, in physics, the slope of a velocity-time graph represents acceleration, providing insights into how an object's speed changes over time. Understanding slopes and perpendicular lines, therefore, extends beyond theoretical mathematics and into practical problem-solving in diverse fields.

In the context of coordinate geometry, the relationship between slopes and perpendicularity is particularly useful when analyzing geometric shapes and figures. For instance, determining whether two lines forming a corner of a shape are perpendicular can help verify if the shape is a rectangle or a square. This involves calculating the slopes of the lines and checking if their product is -1. If it is, the lines are perpendicular, and the corner is a right angle. This principle is also applied in computer graphics and CAD (Computer-Aided Design) software, where precise geometric relationships are crucial for accurate modeling and rendering of objects.

Furthermore, the study of slopes and perpendicular lines lays the foundation for more advanced topics in calculus and linear algebra. In calculus, the derivative of a function at a point gives the slope of the tangent line to the function's graph at that point. Understanding perpendicular slopes helps in finding normal lines, which are perpendicular to the tangent lines and are essential in optimization problems and curve analysis. In linear algebra, the concept of orthogonality, which is a generalization of perpendicularity to higher-dimensional spaces, is built upon the principles of perpendicular lines and their slopes. Thus, a solid grasp of these fundamental concepts is vital for progressing in mathematical studies and applications.

Step-by-Step Solution

Given that the original line has a slope of $-\frac{4}{5}$, we need to find the slope of a line perpendicular to it. As explained earlier, the slope of a perpendicular line is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular line (${m_{\perp}}$) is:

${ m_{\perp} = \frac{5}{4} }$

Now, we need to determine which of the given pairs of points lie on a line with this slope. To do this, we will calculate the slope between each pair of points using the slope formula and check if it equals $\frac{5}{4}$.

Option A: ( $-2, 0$ ) and ( 2, 5 )

Using the slope formula:

${ m = \frac{5 - 0}{2 - (-2)} = \frac{5}{4} }$

The slope between these points is $\frac{5}{4}$, which matches the required slope for a perpendicular line. Therefore, option A is a valid choice.

Option B: ( $-4, 5$ ) and ( 4, $-5$ )

Using the slope formula:

${ m = \frac{-5 - 5}{4 - (-4)} = \frac{-10}{8} = -\frac{5}{4} }$

The slope between these points is $-\frac{5}{4}$, which is not equal to $\frac{5}{4}$. Thus, option B is not a valid choice.

Option C: ( $-3, 4$ ) and ( 2, 0 )

Using the slope formula:

${ m = \frac{0 - 4}{2 - (-3)} = \frac{-4}{5} }$

The slope between these points is $-\frac{4}{5}$, which is not equal to $\frac{5}{4}$. Hence, option C is not a valid choice.

Option D: ( 1, $-1$ ) and ( 6, $-5$ )

Using the slope formula:

${ m = \frac{-5 - (-1)}{6 - 1} = \frac{-4}{5} }$

The slope between these points is $-\frac{4}{5}$, which is also not equal to $\frac{5}{4}$. Therefore, option D is not a valid choice.

Practical Methods and Tips

To solve problems involving slopes and perpendicular lines efficiently, consider the following practical methods and tips:

1. Always start by identifying the slope of the original line. This sets the foundation for finding the slope of the perpendicular line.
2. Remember the negative reciprocal relationship. To find the slope of a perpendicular line, take the negative reciprocal of the original slope. This involves flipping the fraction and changing its sign.
3. Use the slope formula consistently. The slope formula (${m = \frac{y_2 - y_1}{x_2 - x_1}}$) is your primary tool for calculating the slope between two points. Ensure you correctly substitute the coordinates.
4. Simplify fractions. After calculating the slope, simplify the fraction to its lowest terms. This makes it easier to compare slopes and identify relationships.
5. Visually inspect the points if possible. If you have access to a graphing tool, plotting the points can give you a visual sense of the line's slope and direction.
6. Double-check your calculations. Mistakes can easily occur in calculations involving fractions and negative signs. Always review your work to ensure accuracy.
7. Understand the concept behind the formulas. Rote memorization of formulas is not enough; understanding why the formulas work will help you apply them correctly in various situations.
8. Practice regularly. Like any mathematical skill, proficiency in working with slopes and perpendicular lines comes with practice. Work through a variety of problems to build your confidence and understanding.

Conclusion

In summary, determining whether pairs of points lie on a line perpendicular to a given line involves understanding the concept of slopes and the relationship between perpendicular lines. By calculating the slope between given points and comparing it to the negative reciprocal of the original line's slope, we can accurately identify valid options. In the given question, the correct answer is option A, where the points ( $-2, 0$ ) and ( 2, 5 ) form a line with a slope of $\frac{5}{4}$, which is perpendicular to the line with a slope of $-\frac{4}{5}$. This exercise highlights the importance of a solid grasp of coordinate geometry principles and the ability to apply formulas accurately. Understanding these concepts is crucial for success in mathematics and its applications in various fields.

By following the step-by-step solutions and practical methods outlined in this article, students and enthusiasts can confidently tackle problems involving slopes and perpendicular lines. Regular practice and a conceptual understanding of the material will further enhance your problem-solving skills in this area. The ability to work with slopes and perpendicular lines is not only a valuable mathematical skill but also a fundamental tool in many practical applications, making it an essential topic for anyone pursuing studies or careers in science, technology, engineering, and mathematics (STEM) fields.