Understanding Perpendicular Slopes: What Are Negative Reciprocals?
Quick Answer
When two lines are perpendicular, their slopes are negative reciprocals. For a line with slope -2/3, the slope of the perpendicular line is 3/2.
To determine the slope of a line that is perpendicular to another, it’s essential to understand the concept of negative reciprocals. Let's break this down step by step.
When you have a line with a slope, say line e with a slope of -2/3, the slope of a line that is perpendicular to it (let's call it line f) can be found by taking the negative reciprocal of that slope.
### What is a Negative Reciprocal?
A reciprocal of a fraction is simply flipping it. For example, the reciprocal of -2/3 is -3/2. However, since we need the negative reciprocal, we also change the sign, resulting in +3/2. Therefore, the slope of line f should be +3/2.
### Why is This Important?
Understanding slopes and their relationships is crucial in geometry and algebra, particularly when dealing with linear equations. This knowledge is not only important for solving math problems but also for understanding real-world applications such as architecture, engineering, and even computer graphics.
### Example to Illustrate
Imagine you have two roads that cross each other at a right angle. If one road is inclined downwards with a slope of -2/3, the road that intersects it at a right angle will have a slope of +3/2. This means that if you were to graph these two lines on a coordinate plane, you would see that they meet at a right angle, confirming their perpendicularity.
### Common Mistakes
A common mistake is to confuse the negative reciprocal with just the reciprocal. For instance, if you mistakenly thought the slope of line f is -3/2 instead of +3/2, you would be incorrect. Remember, when the original slope is negative, the perpendicular slope must be positive. This is a pivotal concept in geometry that can often lead to confusion, so it’s important to double-check your calculations.
### Quick Tip
If you're ever unsure, remember this simple rule:
1. Take the slope of the original line.
2. Flip the fraction to get the reciprocal.
3. Change the sign to find the negative reciprocal.
With practice, identifying and calculating slopes of perpendicular lines will become second nature, and you'll be well on your way to mastering this essential math concept. Keep practicing, and don’t hesitate to reach out for help if you find yourself stuck!
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