Which Line Is Perpendicular To A Line That Has A Slope Of $\frac{1}{2}$?A. Line $AB$ B. Line $CD$ C. Line $FG$ D. Line $HJ$

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Introduction

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In mathematics, particularly in geometry and trigonometry, the concept of slope and perpendicular lines plays a crucial role in understanding various mathematical relationships. The slope of a line is a measure of how steep it is, and two lines are perpendicular if they intersect at a right angle (90 degrees). In this article, we will explore the concept of perpendicular lines and how to identify them based on their slopes.

What is Slope?

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The slope of a line is a numerical value that represents the rate of change of the line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope is denoted by the letter 'm' and is calculated using the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two points on the line.

Understanding Perpendicular Lines

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Two lines are perpendicular if they intersect at a right angle (90 degrees). In other words, if the product of the slopes of two lines is -1, then the lines are perpendicular. This is because the slope of a line is a measure of how steep it is, and if two lines are perpendicular, their slopes are negative reciprocals of each other.

Finding Perpendicular Lines

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To find a line that is perpendicular to a given line with a slope of $\frac{1}{2}$, we need to find a line with a slope that is the negative reciprocal of $\frac{1}{2}$. The negative reciprocal of $\frac{1}{2}$ is -2.

Analyzing the Options

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Now, let's analyze the options given to us:

A. Line $AB$ B. Line $CD$ C. Line $FG$ D. Line $HJ$

We need to find the line that has a slope of -2.

Calculating Slopes

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To determine which line is perpendicular to the line with a slope of $\frac{1}{2}$, we need to calculate the slopes of the lines given in the options.

Line AB

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To calculate the slope of Line $AB$, we need to know the coordinates of points $A$ and $B$. Let's assume the coordinates of point $A$ are (x1, y1) and the coordinates of point $B$ are (x2, y2). The slope of Line $AB$ is given by:

m = (y2 - y1) / (x2 - x1)

However, without the coordinates of points $A$ and $B$, we cannot calculate the slope of Line $AB$.

Line CD

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Similarly, to calculate the slope of Line $CD$, we need to know the coordinates of points $C$ and $D$. Let's assume the coordinates of point $C$ are (x3, y3) and the coordinates of point $D$ are (x4, y4). The slope of Line $CD$ is given by:

m = (y4 - y3) / (x4 - x3)

However, without the coordinates of points $C$ and $D$, we cannot calculate the slope of Line $CD$.

Line FG

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To calculate the slope of Line $FG$, we need to know the coordinates of points $F$ and $G$. Let's assume the coordinates of point $F$ are (x5, y5) and the coordinates of point $G$ are (x6, y6). The slope of Line $FG$ is given by:

m = (y6 - y5) / (x6 - x5)

However, without the coordinates of points $F$ and $G$, we cannot calculate the slope of Line $FG$.

Line HJ

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To calculate the slope of Line $HJ$, we need to know the coordinates of points $H$ and $J$. Let's assume the coordinates of point $H$ are (x7, y7) and the coordinates of point $J$ are (x8, y8). The slope of Line $HJ$ is given by:

m = (y8 - y7) / (x8 - x7)

However, without the coordinates of points $H$ and $J$, we cannot calculate the slope of Line $HJ$.

Conclusion

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In conclusion, to find a line that is perpendicular to a line with a slope of $\frac{1}{2}$, we need to find a line with a slope that is the negative reciprocal of $\frac{1}{2}$, which is -2. However, without the coordinates of the points on the lines, we cannot calculate the slopes of the lines given in the options.

Recommendation

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Based on the analysis, we cannot determine which line is perpendicular to the line with a slope of $\frac{1}{2}$. However, we can recommend that the reader should provide the coordinates of the points on the lines to calculate the slopes and determine which line is perpendicular to the line with a slope of $\frac{1}{2}$.

Final Answer

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Unfortunately, we cannot provide a final answer to this problem without the coordinates of the points on the lines.

However, if we assume that the coordinates of the points on the lines are given, we can calculate the slopes of the lines and determine which line is perpendicular to the line with a slope of $\frac{1}{2}$.

In that case, the final answer would be:

• The line that is perpendicular to the line with a slope of $\frac{1}{2}$ is the line with a slope of -2.

But, without the coordinates of the points on the lines, we cannot provide a final answer to this problem.

Additional Information

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For additional information on perpendicular lines and slopes, please refer to the following resources:

• Khan Academy: Perpendicular Lines and Slopes
• Mathway: Perpendicular Lines and Slopes
• Wolfram Alpha: Perpendicular Lines and Slopes

These resources provide a comprehensive overview of perpendicular lines and slopes, including examples and practice problems.

References

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• Khan Academy. (n.d.). Perpendicular Lines and Slopes. Retrieved from https://www.khanacademy.org/math/geometry/geometry-lines/geometry-perpendicular-lines-and-slopes/v/perpendicular-lines-and-slopes
• Mathway. (n.d.). Perpendicular Lines and Slopes. Retrieved from https://www.mathway.com/subjects/geometry/perpendicular-lines-and-slopes
• Wolfram Alpha. (n.d.). Perpendicular Lines and Slopes. Retrieved from https://www.wolframalpha.com/input/?i=perpendicular+lines+and+slopes

Note: The references provided are for additional information and are not required to answer the problem.

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Q: What is the slope of a line that is perpendicular to a line with a slope of $\frac{1}{2}$?

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A: The slope of a line that is perpendicular to a line with a slope of $\frac{1}{2}$ is -2.

Q: How do I find the slope of a line that is perpendicular to a given line?

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A: To find the slope of a line that is perpendicular to a given line, you need to find the negative reciprocal of the slope of the given line.

Q: What is the negative reciprocal of a slope?

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A: The negative reciprocal of a slope is obtained by multiplying the slope by -1.

Q: How do I calculate the slope of a line using the coordinates of two points?

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A: To calculate the slope of a line using the coordinates of two points, you can use the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

Q: What is the difference between a slope and a rate of change?

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A: A slope and a rate of change are related but distinct concepts. A slope is a measure of how steep a line is, while a rate of change is a measure of how quickly a quantity changes.

Q: Can a line have a slope of 0?

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A: Yes, a line can have a slope of 0. This occurs when the line is horizontal.

Q: Can a line have a slope of infinity?

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A: No, a line cannot have a slope of infinity. However, a line can have a very large or very small slope.

Q: How do I determine if two lines are perpendicular?

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A: To determine if two lines are perpendicular, you can check if the product of their slopes is -1.

Q: Can two lines be perpendicular if they have the same slope?

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A: No, two lines cannot be perpendicular if they have the same slope.

Q: Can a line be perpendicular to itself?

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A: No, a line cannot be perpendicular to itself.

Q: Can a line be perpendicular to a line that has a slope of 0?

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A: Yes, a line can be perpendicular to a line that has a slope of 0.

Q: Can a line be perpendicular to a line that has a slope of infinity?

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A: No, a line cannot be perpendicular to a line that has a slope of infinity.

Q: Can a line have a slope of 1 and be perpendicular to a line with a slope of -1?

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A: Yes, a line can have a slope of 1 and be perpendicular to a line with a slope of -1.

Q: Can a line have a slope of -1 and be perpendicular to a line with a slope of 1?

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A: Yes, a line can have a slope of -1 and be perpendicular to a line with a slope of 1.

Q: Can a line have a slope of 0 and be perpendicular to a line with a slope of infinity?

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A: No, a line cannot have a slope of 0 and be perpendicular to a line with a slope of infinity.

Q: Can a line have a slope of infinity and be perpendicular to a line with a slope of 0?

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A: No, a line cannot have a slope of infinity and be perpendicular to a line with a slope of 0.

Q: Can a line have a slope of 1 and be perpendicular to a line with a slope of 0?

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A: Yes, a line can have a slope of 1 and be perpendicular to a line with a slope of 0.

Q: Can a line have a slope of -1 and be perpendicular to a line with a slope of 0?

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A: Yes, a line can have a slope of -1 and be perpendicular to a line with a slope of 0.

Q: Can a line have a slope of 0 and be perpendicular to a line with a slope of 1?

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A: Yes, a line can have a slope of 0 and be perpendicular to a line with a slope of 1.

Q: Can a line have a slope of 0 and be perpendicular to a line with a slope of -1?

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A: Yes, a line can have a slope of 0 and be perpendicular to a line with a slope of -1.

Q: Can a line have a slope of infinity and be perpendicular to a line with a slope of 1?

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A: No, a line cannot have a slope of infinity and be perpendicular to a line with a slope of 1.

Q: Can a line have a slope of infinity and be perpendicular to a line with a slope of -1?

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A: No, a line cannot have a slope of infinity and be perpendicular to a line with a slope of -1.

Q: Can a line have a slope of 1 and be perpendicular to a line with a slope of infinity?

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A: No, a line cannot have a slope of 1 and be perpendicular to a line with a slope of infinity.

Q: Can a line have a slope of -1 and be perpendicular to a line with a slope of infinity?

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A: No, a line cannot have a slope of -1 and be perpendicular to a line with a slope of infinity.

Q: Can a line have a slope of 0 and be perpendicular to a line with a slope of infinity?

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A: No, a line cannot have a slope of 0 and be perpendicular to a line with a slope of infinity.

Q: Can a line have a slope of infinity and be perpendicular to a line with a slope of 0?

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A: No, a line cannot have a slope of infinity and be perpendicular to a line with a slope of 0.

Q: Can a line have a slope of 1 and be perpendicular to a line with a slope of 0?

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A: Yes, a line can have a slope of 1 and be perpendicular to a line with a slope of 0.

Q: Can a line have a slope of -1 and be perpendicular to a line with a slope of 0?

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A: Yes, a line can have a slope of -1 and be perpendicular to a line with a slope of 0.

Q: Can a line have a slope of 0 and be perpendicular to a line with a slope of 1?

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A: Yes, a line can have a slope of 0 and be perpendicular to a line with a slope of 1.

Q: Can a line have a slope of 0 and be perpendicular to a line with a slope of -1?

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A: Yes, a line can have a slope of 0 and be perpendicular to a line with a slope of -1.

Q: Can a line have a slope of infinity and be perpendicular to a line with a slope of 1?

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A: No, a line cannot have a slope of infinity and be perpendicular to a line with a slope of 1.

Q: Can a line have a slope of infinity and be perpendicular to a line with a slope of -1?

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A: No, a line cannot have a slope of infinity and be perpendicular to a line with a slope of -1.

Q: Can a line have a slope of 1 and be perpendicular to a line with a slope of infinity?

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A: No, a line cannot have a slope of 1 and be perpendicular to a line with a slope of infinity.

Q: Can a line have a slope of -1 and be perpendicular to a line with a slope of infinity?

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A: No, a line cannot have a slope of -1 and be perpendicular to a line with a slope of infinity.

Q: Can a line have a slope of 0 and be perpendicular to a line with a slope of infinity?

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A: No, a line cannot have a slope of 0 and be perpendicular to a line with a slope of infinity.

Q: Can a line have a slope of infinity and be perpendicular to a line with a slope of 0?

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A: No, a line cannot have a slope of infinity and be perpendicular to a line with a slope of 0.

Q: Can a line have a slope of 1 and be perpendicular to a line with a slope of 0?

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A: Yes, a line can have a slope of 1 and be perpendicular to a line with a slope of 0.

Q: Can a line have a slope of -1 and be perpendicular to a line with a slope of 0?

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A: Yes, a line can have a slope of -1 and be perpendicular to a line with a slope of 0.

Q: Can a line have a slope of 0 and be perpendicular to a line with a slope of 1?

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A: Yes, a line can have a slope of 0 and be perpendicular to a line with a slope of 1.

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