A Line Has A Slope Of − 4 5 -\frac{4}{5} − 5 4 ​ . Which Ordered Pairs Could Be Points On A Line That Is Perpendicular To This Line? Select Two Options.A. ( − 2 , 0 (-2,0 ( − 2 , 0 ] And ( 2 , 5 (2,5 ( 2 , 5 ]B. ( − 4 , 5 (-4,5 ( − 4 , 5 ] And ( 4 , − 5 (4,-5 ( 4 , − 5 ]C. ( − 3 , 4 (-3,4 ( − 3 , 4 ]

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Understanding Slope and Perpendicular Lines

When dealing with lines and their slopes, it's essential to understand the concept of perpendicular lines. Two lines are perpendicular if their slopes are negative reciprocals of each other. In other words, if the slope of one line is $m$, then the slope of a line perpendicular to it is $-\frac{1}{m}$.

The Given Line and Its Slope

The given line has a slope of $-\frac{4}{5}$. To find the slope of a line perpendicular to this line, we need to find the negative reciprocal of $-\frac{4}{5}$. The negative reciprocal of a fraction is obtained by flipping the fraction and changing the sign. Therefore, the slope of a line perpendicular to the given line is $\frac{5}{4}$.

Finding Perpendicular Lines

Now that we know the slope of a line perpendicular to the given line, we can use this information to find the ordered pairs that could be points on such a line. We will consider the options provided and determine which ones satisfy the condition of having a slope of $\frac{5}{4}$.

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

To determine if this option is correct, we need to find the slope of the line passing through the points $(-2,0)$ and $(2,5)$. The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

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

Substituting the given points into this formula, we get:

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

Since the slope of the line passing through the points $(-2,0)$ and $(2,5)$ is $\frac{5}{4}$, which is the same as the slope of a line perpendicular to the given line, this option is correct.

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

To determine if this option is correct, we need to find the slope of the line passing through the points $(-4,5)$ and $(4,-5)$. Using the same formula as before, we get:

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

Since the slope of the line passing through the points $(-4,5)$ and $(4,-5)$ is $-\frac{5}{4}$, which is not the same as the slope of a line perpendicular to the given line, this option is incorrect.

Option C: $(-3,4)$

This option only provides one point, $(-3,4)$. To determine if this option is correct, we need to find the slope of the line passing through this point and another point on the line. However, since only one point is provided, we cannot determine the slope of the line passing through this point and another point on the line.

Conclusion

Based on the analysis above, the correct options are A and C. Option A provides two points, $(-2,0)$ and $(2,5)$, which satisfy the condition of having a slope of $\frac{5}{4}$. Option C provides one point, $(-3,4)$, which could be a point on a line with a slope of $\frac{5}{4}$.

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Understanding Slope and Perpendicular Lines

When dealing with lines and their slopes, it's essential to understand the concept of perpendicular lines. Two lines are perpendicular if their slopes are negative reciprocals of each other. In other words, if the slope of one line is $m$, then the slope of a line perpendicular to it is $-\frac{1}{m}$.

The Given Line and Its Slope

The given line has a slope of $-\frac{4}{5}$. To find the slope of a line perpendicular to this line, we need to find the negative reciprocal of $-\frac{4}{5}$. The negative reciprocal of a fraction is obtained by flipping the fraction and changing the sign. Therefore, the slope of a line perpendicular to the given line is $\frac{5}{4}$.

Finding Perpendicular Lines

Now that we know the slope of a line perpendicular to the given line, we can use this information to find the ordered pairs that could be points on such a line. We will consider the options provided and determine which ones satisfy the condition of having a slope of $\frac{5}{4}$.

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

To determine if this option is correct, we need to find the slope of the line passing through the points $(-2,0)$ and $(2,5)$. The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

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

Substituting the given points into this formula, we get:

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

Since the slope of the line passing through the points $(-2,0)$ and $(2,5)$ is $\frac{5}{4}$, which is the same as the slope of a line perpendicular to the given line, this option is correct.

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

To determine if this option is correct, we need to find the slope of the line passing through the points $(-4,5)$ and $(4,-5)$. Using the same formula as before, we get:

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

Since the slope of the line passing through the points $(-4,5)$ and $(4,-5)$ is $-\frac{5}{4}$, which is not the same as the slope of a line perpendicular to the given line, this option is incorrect.

Option C: $(-3,4)$

This option only provides one point, $(-3,4)$. To determine if this option is correct, we need to find the slope of the line passing through this point and another point on the line. However, since only one point is provided, we cannot determine the slope of the line passing through this point and another point on the line.

Conclusion

Based on the analysis above, the correct options are A and C. Option A provides two points, $(-2,0)$ and $(2,5)$, which satisfy the condition of having a slope of $\frac{5}{4}$. Option C provides one point, $(-3,4)$, which could be a point on a line with a slope of $\frac{5}{4}$.

Q&A

Q: What is the slope of a line perpendicular to a line with a slope of $-\frac{4}{5}$?

A: The slope of a line perpendicular to a line with a slope of $-\frac{4}{5}$ is $\frac{5}{4}$.

Q: How do you find the slope of a line passing through two points?

A: The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

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

Q: What is the condition for two lines to be perpendicular?

A: Two lines are perpendicular if their slopes are negative reciprocals of each other. In other words, if the slope of one line is $m$, then the slope of a line perpendicular to it is $-\frac{1}{m}$.

Q: Can a line have a slope of $\frac{5}{4}$?

A: Yes, a line can have a slope of $\frac{5}{4}$. This is the slope of a line perpendicular to a line with a slope of $-\frac{4}{5}$.

Q: What is the relationship between the slope of a line and the slope of a line perpendicular to it?

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

Q: Can a line have a slope of $-\frac{5}{4}$?

A: Yes, a line can have a slope of $-\frac{5}{4}$. This is the slope of a line passing through the points $(-4,5)$ and $(4,-5)$.

Q: What is the significance of the slope of a line in determining the relationship between two lines?

A: The slope of a line is crucial in determining the relationship between two lines. If the slopes of two lines are equal, then the lines are parallel. If the slopes of two lines are negative reciprocals of each other, then the lines are perpendicular.

Q: Can a line have a slope of $0$?

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

Q: Can a line have a slope of $\infty$?

A: Yes, a line can have a slope of $\infty$. This is the case when the line is vertical.

Q: What is the relationship between the slope of a line and the angle it makes with the x-axis?

A: The slope of a line is equal to the tangent of the angle it makes with the x-axis.

Q: Can a line have a slope that is not a rational number?

A: Yes, a line can have a slope that is not a rational number. This is the case when the slope is an irrational number.

Q: What is the significance of the slope of a line in determining the equation of the line?

A: The slope of a line is crucial in determining the equation of the line. The equation of a line can be written in the form $y = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept.

Q: Can a line have a slope that is not a real number?

A: No, a line cannot have a slope that is not a real number. The slope of a line is always a real number.

Q: What is the relationship between the slope of a line and the slope of a line parallel to it?

A: The slope of a line parallel to a line with a slope of $m$ is also $m$.

Q: Can a line have a slope that is greater than $1$?

A: Yes, a line can have a slope that is greater than $1$. This is the case when the line is steeper than a $45^\circ$ angle.

Q: Can a line have a slope that is less than $-1$?

A: Yes, a line can have a slope that is less than $-1$. This is the case when the line is steeper than a $-45^\circ$ angle.

Q: What is the significance of the slope of a line in determining the direction of the line?

A: The slope of a line is crucial in determining the direction of the line. A line with a positive slope is increasing, while a line with a negative slope is decreasing.

Q: Can a line have a slope that is equal to $1$?

A: Yes, a line can have a slope that is equal to $1$. This is the case when the line is at a $45^\circ$ angle.

Q: Can a line have a slope that is equal to $-1$?

A: Yes, a line can have a slope that is equal to $-1$. This is the case when the line is at a $-45^\circ$ angle.

Q: What is the relationship between the slope of a line and the slope of a line that is a reflection of it?

A: The slope of a line that is a reflection of a line with a slope of $m$ is $-m$.

Q: Can a line have a slope that is equal to $0$ and a slope that is equal to $\infty$ at the same time?

A: No, a line cannot have a slope that is equal to $0$ and a slope that is equal to $\infty$ at the same time. A line with a slope of $0$ is horizontal, while a line with a slope of $\infty$ is vertical.

Q: What is the significance of the slope of a line in determining the equation of the line in a three-dimensional space?

A: The slope of a line in a three-dimensional space is crucial in determining the equation of the line. The equation of a line in three-dimensional