Question
The location of runway A for a new airport is shown on the coordinate plane below.
A line labeled 'Runway A' intercepting x axis at -4 and y-axis at 3.
Two additional runways are planned. Each will pass through the point
on the coordinate plane.
Runway B is to be parallel to runway A.
Runway C is to be perpendicular to runway A.
Which equations could describe either of those runways? Choose all that are correct.
A.
B.
C.
D.
E.
F.
A line labeled 'Runway A' intercepting x axis at -4 and y-axis at 3.
Two additional runways are planned. Each will pass through the point
on the coordinate plane.
Runway B is to be parallel to runway A.
Runway C is to be perpendicular to runway A.
Which equations could describe either of those runways? Choose all that are correct.
A.
B.
C.
D.
E.
F.
Answers
GPT-4o mini
To find the equations of Runways B and C, we first need the equation of Runway A. From the information given:
- Runway A intercepts the x-axis at -4 and y-axis at 3.
Using the intercepts, we can find the slope of Runway A. The slope \( m \) can be calculated using the formula:
\[
m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Choosing the points (-4, 0) and (0, 3):
\[
m = \frac{3 - 0}{0 - (-4)} = \frac{3}{4}
\]
So the slope of Runway A is \( \frac{3}{4} \). The equation of a line in slope-intercept form is:
\[
y = mx + b
\]
Here, b is the y-intercept, which is 3 for Runway A:
\[
y = \frac{3}{4}x + 3
\]
### Runway B (Parallel to Runway A)
The slope of any line parallel to Runway A will be the same, \( m = \frac{3}{4} \). Thus, the general equation for Runway B would be:
\[
y = \frac{3}{4}x + b
\]
where \( b \) is some value that you can choose.
### Runway C (Perpendicular to Runway A)
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. The negative reciprocal of \( \frac{3}{4} \) is \( -\frac{4}{3} \). Therefore, the equation of Runway C could be written as:
\[
y = -\frac{4}{3}x + b
\]
where \( b \) is again some value that can be chosen based on the specific point it should pass through.
### Conclusion
Without specific options A, B, C, D, E, or F presented as equations, I can't choose specific answers. However, any equation with a slope of \( \frac{3}{4} \) for Runway B and a slope of \( -\frac{4}{3} \) for Runway C would be valid. If you provide those options, I can help you pick the correct ones.
- Runway A intercepts the x-axis at -4 and y-axis at 3.
Using the intercepts, we can find the slope of Runway A. The slope \( m \) can be calculated using the formula:
\[
m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Choosing the points (-4, 0) and (0, 3):
\[
m = \frac{3 - 0}{0 - (-4)} = \frac{3}{4}
\]
So the slope of Runway A is \( \frac{3}{4} \). The equation of a line in slope-intercept form is:
\[
y = mx + b
\]
Here, b is the y-intercept, which is 3 for Runway A:
\[
y = \frac{3}{4}x + 3
\]
### Runway B (Parallel to Runway A)
The slope of any line parallel to Runway A will be the same, \( m = \frac{3}{4} \). Thus, the general equation for Runway B would be:
\[
y = \frac{3}{4}x + b
\]
where \( b \) is some value that you can choose.
### Runway C (Perpendicular to Runway A)
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. The negative reciprocal of \( \frac{3}{4} \) is \( -\frac{4}{3} \). Therefore, the equation of Runway C could be written as:
\[
y = -\frac{4}{3}x + b
\]
where \( b \) is again some value that can be chosen based on the specific point it should pass through.
### Conclusion
Without specific options A, B, C, D, E, or F presented as equations, I can't choose specific answers. However, any equation with a slope of \( \frac{3}{4} \) for Runway B and a slope of \( -\frac{4}{3} \) for Runway C would be valid. If you provide those options, I can help you pick the correct ones.