Understanding Slope and Perpendicular Lines

Oct 13, 2025 by miodeya.com 94 views

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 multiply to -1. In other words, if the slope of one line is m, then the slope of a line perpendicular to it is -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 multiply the slope by -1. Therefore, the slope of a line perpendicular to the given line is $\frac{5}{4}$ .

Finding Perpendicular Lines

To find the ordered pairs that could be points on a line perpendicular to the given line, we need to find the equation of the line with a slope of $\frac{5}{4}$ . However, we are not given any points on the line, so we will use the point-slope form of a line, which is given by:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope.

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

Let's consider the first option, (-2,0) and (2,5). To determine if these points are on a line with a slope of $\frac{5}{4}$ , we can use the point-slope form of a line. We can choose either point as (x1, y1) and use the slope to find the equation of the line.

Using the Point (-2,0)

If we choose the point (-2,0) as (x1, y1), we can plug in the values into the point-slope form of a line:

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

Simplifying the equation, we get:

y = $\frac{5}{4}$ (x + 2)

Using the Point (2,5)

If we choose the point (2,5) as (x1, y1), we can plug in the values into the point-slope form of a line:

y - 5 = $\frac{5}{4}$ (x - 2)

Simplifying the equation, we get:

y = $\frac{5}{4}$ (x - 2) + 5

Checking the Slope

To check if the slope of the line is indeed $\frac{5}{4}$ , we can rewrite the equation in slope-intercept form, which is given by:

y = mx + b

where m is the slope and b is the y-intercept.

For the first equation, we get:

y = $\frac{5}{4}$ (x + 2)

y = $\frac{5}{4}$ x + $\frac{5}{2}$

The slope of this line is indeed $\frac{5}{4}$ .

For the second equation, we get:

y = $\frac{5}{4}$ (x - 2) + 5

y = $\frac{5}{4}$ x - $\frac{5}{2}$ + 5

y = $\frac{5}{4}$ x + $\frac{5}{2}$

The slope of this line is also $\frac{5}{4}$ .

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

Let's consider the second option, (-4,5) and (4,-5). To determine if these points are on a line with a slope of $\frac{5}{4}$ , we can use the point-slope form of a line. We can choose either point as (x1, y1) and use the slope to find the equation of the line.

Using the Point (-4,5)

If we choose the point (-4,5) as (x1, y1), we can plug in the values into the point-slope form of a line:

y - 5 = $\frac{5}{4}$ (x - (-4))

Simplifying the equation, we get:

y = $\frac{5}{4}$ (x + 4) + 5

Using the Point (4,-5)

If we choose the point (4,-5) as (x1, y1), we can plug in the values into the point-slope form of a line:

y - (-5) = $\frac{5}{4}$ (x - 4)

Simplifying the equation, we get:

y = $\frac{5}{4}$ (x - 4) - 5

Checking the Slope

To check if the slope of the line is indeed $\frac{5}{4}$ , we can rewrite the equation in slope-intercept form, which is given by:

y = mx + b

where m is the slope and b is the y-intercept.

For the first equation, we get:

y = $\frac{5}{4}$ (x + 4) + 5

y = $\frac{5}{4}$ x + 5 + 5

y = $\frac{5}{4}$ x + 10

The slope of this line is indeed $\frac{5}{4}$ .

For the second equation, we get:

y = $\frac{5}{4}$ (x - 4) - 5

y = $\frac{5}{4}$ x - 5 - 5

y = $\frac{5}{4}$ x - 10

The slope of this line is also $\frac{5}{4}$ .

Conclusion

Based on the calculations, we can conclude that both options A and B contain points that are on a line with a slope of $\frac{5}{4}$ , which is perpendicular to the given line with a slope of $-\frac{4}{5}$ .

Q&A: Understanding Slope and Perpendicular Lines

Q: What is the concept of perpendicular lines?

A: Two lines are perpendicular if their slopes multiply to -1. In other words, if the slope of one line is m, then the slope of a line perpendicular to it is -1/m.

Q: How do you 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 multiply the slope of the given line by -1.

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 equation of a line with a given slope and a point on the line?

A: You can use the point-slope form of a line, which is given by:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope.

Q: How do you check if the slope of a line is indeed the given slope?

A: You can rewrite the equation in slope-intercept form, which is given by:

y = mx + b

where m is the slope and b is the y-intercept.

Q: What are the two options that contain points that are on a line with a slope of $\frac{5}{4}$ ?

A: The two options are:

A. $(-2,0)$ and $(2,5)$ B. $(-4,5)$ and $(4,-5)$

Q: How do you determine if a point is on a line with a given slope?

A: You can use the point-slope form of a line and plug in the values of the point and the slope to see if the equation is true.

Q: What is the importance of understanding slope and perpendicular lines?

A: Understanding slope and perpendicular lines is important in mathematics and real-world applications, such as architecture, engineering, and physics.

Q: How do you find the equation of a line with a given slope and two points on the line?

A: You can use the two-point form of a line, which is given by:

y - y1 = m(x - x1)

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

Q: What are some real-world applications of understanding slope and perpendicular lines?

A: Some real-world applications include:

Building design and architecture
Engineering and construction
Physics and mechanics
Computer graphics and game development

Q: How do you use technology to graph and analyze lines with given slopes?

A: You can use graphing calculators, computer software, or online tools to graph and analyze lines with given slopes.

Q: What are some common mistakes to avoid when working with slope and perpendicular lines?

A: Some common mistakes to avoid include:

Confusing the slope with the y-intercept
Not checking if the slope is indeed the given slope
Not using the correct formula for the equation of a line
Not considering the context and real-world applications of the problem.