Which Of The Following Lines Is Perpendicular To The Line $y = -2x + 5$?Choose The Correct Answer Below.A. $y = -\frac{1}{2}x + 3$ B. $y = \frac{1}{2}x - 3$ C. $y = 2x + 1$ D. $y = -2x - \frac{1}{5}$ E.

Which of the Following Lines is Perpendicular to the Line $y = -2x + 5$?

In mathematics, particularly in geometry and algebra, the concept of perpendicular lines plays a crucial role in understanding various mathematical concepts. Perpendicular lines are lines that intersect at a right angle, which is 90 degrees. In this article, we will explore which of the given lines is perpendicular to the line $y = -2x + 5$. To determine this, we need to understand the concept of slope and how it relates to perpendicular lines.

The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope of a line can be represented by the formula:

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

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

Slope of the Given Line

The given line is $y = -2x + 5$. To find the slope of this line, we can rewrite it in the slope-intercept form, which is $y = mx + b$, where m is the slope and b is the y-intercept. In this case, the slope (m) is -2.

Perpendicular lines have slopes that are negative reciprocals of each other. This means that if the slope of one line is m, the slope of its perpendicular line is -1/m.

Finding the Perpendicular Line

To find the perpendicular line to the given line $y = -2x + 5$, we need to find a line with a slope that is the negative reciprocal of -2. The negative reciprocal of -2 is 1/2.

Analyzing the Options

Now that we know the slope of the perpendicular line is 1/2, we can analyze the given options to determine which one is perpendicular to the line $y = -2x + 5$.

Option A: $y = -\frac{1}{2}x + 3$

The slope of this line is -1/2, which is not the negative reciprocal of -2. Therefore, this line is not perpendicular to the given line.

Option B: $y = \frac{1}{2}x - 3$

The slope of this line is 1/2, which is the negative reciprocal of -2. Therefore, this line is perpendicular to the given line.

Option C: $y = 2x + 1$

The slope of this line is 2, which is not the negative reciprocal of -2. Therefore, this line is not perpendicular to the given line.

Option D: $y = -2x - \frac{1}{5}$

The slope of this line is -2, which is the same as the slope of the given line. Therefore, this line is not perpendicular to the given line.

Option E: Not Provided

Since option E is not provided, we cannot analyze it.

In conclusion, the line that is perpendicular to the line $y = -2x + 5$ is $y = \frac{1}{2}x - 3$. This line has a slope of 1/2, which is the negative reciprocal of -2, the slope of the given line.
Frequently Asked Questions (FAQs) About Perpendicular Lines

In our previous article, we discussed how to determine which line is perpendicular to the line $y = -2x + 5$. In this article, we will answer some frequently asked questions about perpendicular lines to help you better understand this concept.

Q: What is the definition of perpendicular lines?

A: Perpendicular lines are lines that intersect at a right angle, which is 90 degrees. In other words, if two lines are perpendicular, they form a right angle when they intersect.

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

A: To determine if two lines are perpendicular, you need to find the slopes of the two lines. If the slopes are negative reciprocals of each other, then the lines are perpendicular.

Q: What is the negative reciprocal of a slope?

A: The negative reciprocal of a slope is the number that, when multiplied by the original slope, gives -1. For example, the negative reciprocal of 2 is -1/2, and the negative reciprocal of -2 is 1/2.

Q: How do I find the negative reciprocal of a slope?

A: To find the negative reciprocal of a slope, you can follow these steps:

1. Take the original slope and change its sign (i.e., make it positive if it's negative, and vice versa).
2. Take the reciprocal of the slope (i.e., flip the fraction).
3. The result is the negative reciprocal of the original slope.

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

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

Q: Can a line be perpendicular to itself?

A: No, a line cannot be perpendicular to itself. Perpendicular lines are lines that intersect at a right angle, and a line cannot intersect itself at a right angle.

Q: Can a line be perpendicular to a line that is parallel to it?

A: No, a line cannot be perpendicular to a line that is parallel to it. If two lines are parallel, they have the same slope, and therefore, they cannot be perpendicular.

Q: Can a line be perpendicular to a line that is the same as it?

A: No, a line cannot be perpendicular to a line that is the same as it. If two lines are the same, they have the same slope, and therefore, they cannot be perpendicular.

In conclusion, perpendicular lines are lines that intersect at a right angle, and they have slopes that are negative reciprocals of each other. We hope that this article has helped you better understand this concept and has answered some of the frequently asked questions about perpendicular lines.