Perpendicular Lines Identify Ordered Pairs
When delving into the fascinating world of linear equations and geometry, the concept of perpendicular lines holds significant importance. In essence, two lines are deemed perpendicular if they intersect at a right angle (90 degrees). This fundamental geometric relationship has a profound impact on their slopes. Our exploration today focuses on identifying ordered pairs that could lie on a line perpendicular to a given line with a slope of -4/5. This exercise not only reinforces our understanding of slopes and perpendicularity but also hones our analytical skills in coordinate geometry.
The Relationship Between Slopes of Perpendicular Lines
At the heart of determining perpendicularity lies the relationship between the slopes of the lines involved. The slope of a line is a measure of its steepness and direction, often represented as "rise over run." In mathematical terms, the slope (m) is calculated as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line: m = (y2 - y1) / (x2 - x1).
Now, for two lines to be perpendicular, their slopes must satisfy a specific condition: they must be 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. The negative reciprocal relationship is crucial because it ensures that the lines intersect at a right angle. When you multiply the slopes of two perpendicular lines, the result is always -1. This property provides a powerful tool for verifying whether two lines are indeed perpendicular.
In our given problem, the line has a slope of -4/5. To find the slope of a line perpendicular to it, we need to calculate the negative reciprocal. First, we take the reciprocal of -4/5, which is -5/4. Then, we negate it, resulting in 5/4. Therefore, any line perpendicular to the given line must have a slope of 5/4. This crucial piece of information guides us in identifying the correct ordered pairs from the options provided.
Analyzing the Options
Now that we know the slope of any line perpendicular to the given line must be 5/4, we can analyze the provided ordered pairs to determine which ones satisfy this condition. We will use the slope formula, m = (y2 - y1) / (x2 - x1), to calculate the slope between each pair of points. If the calculated slope matches 5/4, then the ordered pairs could lie on a line perpendicular to the given line.
Option A: (-2, 0) and (2, 5)
Let's calculate the slope between these points: m = (5 - 0) / (2 - (-2)) = 5 / 4. The slope is indeed 5/4, which matches the required slope for a perpendicular line. Therefore, option A is a potential answer.
Option B: (-4, 5) and (4, -5)
Calculating the slope for this pair: m = (-5 - 5) / (4 - (-4)) = -10 / 8 = -5/4. This slope is the negative reciprocal of the perpendicular slope we need, not the perpendicular slope itself. Thus, option B is not a correct choice.
Option C: (-3, 4) and (2, 0)
For this option, the slope is: m = (0 - 4) / (2 - (-3)) = -4 / 5. This slope is the same as the slope of the original line, indicating that these points would lie on a line parallel to the given line, not perpendicular. Hence, option C is incorrect.
Option D: (1, -1) and (6, -5)
Calculating the slope: m = (-5 - (-1)) / (6 - 1) = -4 / 5. Similar to option C, this slope matches the original line's slope, indicating parallelism, not perpendicularity. Therefore, option D is not a valid answer.
Selecting the Correct Options
Based on our analysis, only option A has a slope of 5/4, which is the negative reciprocal of the given slope (-4/5). This confirms that the points in option A could lie on a line perpendicular to the original line. However, the question asks us to select two options, implying that there must be another correct answer. Let's re-examine our calculations and reasoning to ensure we haven't missed anything.
Upon closer inspection, we realize there was an oversight. We correctly identified that option A has a slope of 5/4, which makes it a valid answer. However, we prematurely dismissed option B because it had a slope of -5/4. While -5/4 is not the perpendicular slope we were looking for (5/4), it's essential to remember the question asks for ordered pairs that could be points on a line that is perpendicular to this line. The crucial phrase here is could be. A line with a slope of 5/4 is perpendicular to the line with a slope of -4/5, but so is a line with a slope of 5/4. We need to find ordered pairs that fit the perpendicular relationship, not necessarily the line with the inverse positive slope.
Option A gives us a slope of 5/4, which is the correct perpendicular slope. Now let's think about Option B. Option B gives us a slope of -5/4. However, if we look for a set of points that when connected make a slope that is the negative inverse of -4/5, it should be 5/4.
Therefore, after careful consideration and re-evaluation, we can confidently select two options that satisfy the condition of perpendicularity:
- Option A: (-2, 0) and (2, 5)
- There appears to be a mistake in the options provided. Only Option A results in the correct perpendicular slope of 5/4.
Key Takeaways
This exercise highlights several key concepts in coordinate geometry:
- Perpendicular lines intersect at a right angle (90 degrees).
- The slopes of perpendicular lines are negative reciprocals of each other.
- The slope formula, m = (y2 - y1) / (x2 - x1), is used to calculate the slope between two points.
- Careful analysis and attention to detail are crucial in solving mathematical problems.
By understanding these principles, we can confidently tackle a wide range of problems involving linear equations and geometric relationships. Remember, practice and a thorough understanding of fundamental concepts are the keys to success in mathematics.
In conclusion, while option A definitively represents a pair of points on a line perpendicular to the given line, Option B, C and D do not satisfy the condition. It is vital to remember that the relationship between slopes of perpendicular lines is fundamental in coordinate geometry and mastering it is crucial for solving related problems. This exploration underscores the importance of meticulous calculation and a comprehensive understanding of the underlying principles in mathematical problem-solving.
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"Given a line with a slope of -4/5, identify two ordered pairs that could lie on a perpendicular line."
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Perpendicular Lines Identify Ordered Pairs on a Perpendicular Line