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 ]

$$

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 its 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. To do this, we need to use the point-slope form of a line, which is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

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

Let's consider the first option, which is $(-2,0)$ and $(2,5)$. To determine if these points lie on a line with a slope of $\frac{5}{4}$, we can use the slope formula, which is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$. Plugging in the values of the two points, we get:

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

Since the slope of the line passing through the two points is $\frac{5}{4}$, which is the same as the slope of a line perpendicular to the given line, we can conclude that the points $(-2,0)$ and $(2,5)$ lie on a line that is perpendicular to the given line.

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

Now, let's consider the second option, which is $(-4,5)$ and $(4,-5)$. To determine if these points lie on a line with a slope of $\frac{5}{4}$, we can use the slope formula again:

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

Since the slope of the line passing through the two points is $-\frac{5}{4}$, which is not the same as the slope of a line perpendicular to the given line, we can conclude that the points $(-4,5)$ and $(4,-5)$ do not lie on a line that is perpendicular to the given line.

Conclusion

In conclusion, the only option that satisfies the condition of having points on a line that is perpendicular to the given line is option A, which is $(-2,0)$ and $(2,5)$.

Final Answer

The final answer is option A, which is $(-2,0)$ and $(2,5)$.

$$

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 $-\frac{1}{m}$.

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 I find the slope of a line perpendicular to a given line?

A: To find the slope of a line perpendicular to a given line, you need to find the negative reciprocal of the slope of the given line. The negative reciprocal of a fraction is obtained by flipping the fraction and changing its sign.

Q: What is the point-slope form of a line?

A: The point-slope form of a line is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Q: How do I determine if two points lie on a line with a given slope?

A: To determine if two points lie on a line with a given slope, you can use the slope formula, which is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$. If the slope of the line passing through the two points is equal to the given slope, then the points lie on the line.

Q: What is the relationship between the slopes of perpendicular lines?

A: The slopes of perpendicular lines 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 more than one perpendicular line?

A: Yes, a line can have more than one perpendicular line. In fact, for every line, there are infinitely many lines that are perpendicular to it.

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

A: To find the equation of a line that is perpendicular to a given line, you need to find the slope of the perpendicular line and then use the point-slope form of a line to write the equation of the line.

Q: What is the significance of perpendicular lines in real-world applications?

A: Perpendicular lines have many real-world applications, such as in architecture, engineering, and physics. For example, in building design, perpendicular lines are used to create stable and balanced structures.

Q: Can perpendicular lines be parallel?

A: No, perpendicular lines cannot be parallel. By definition, perpendicular lines intersect at a single point, whereas parallel lines never intersect.

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

A: To determine if two lines are perpendicular, you can check if their slopes are negative reciprocals of each other. If the slopes are negative reciprocals, then the lines are perpendicular.

Q: What is the relationship between the slopes of parallel lines?

A: The slopes of parallel lines are equal. In other words, if the slope of one line is $m$, then the slope of a line parallel to it is also $m$.

Q: Can a line be both parallel and perpendicular to another line?

A: No, a line cannot be both parallel and perpendicular to another line. These two properties are mutually exclusive.

Q: How do I find the equation of a line that is parallel to a given line?

A: To find the equation of a line that is parallel to a given line, you need to find the slope of the parallel line and then use the point-slope form of a line to write the equation of the line.

Q: What is the significance of parallel lines in real-world applications?

A: Parallel lines have many real-world applications, such as in architecture, engineering, and physics. For example, in building design, parallel lines are used to create symmetrical and balanced structures.

Q: Can parallel lines be perpendicular?

A: No, parallel lines cannot be perpendicular. By definition, parallel lines never intersect, whereas perpendicular lines intersect at a single point.

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

A: To determine if two lines are parallel, you can check if their slopes are equal. If the slopes are equal, then the lines are parallel.

Q: What is the relationship between the slopes of perpendicular and parallel lines?

A: The slopes of perpendicular lines are negative reciprocals of each other, whereas the slopes of parallel lines are equal.