Question
Question
The location of runway A for a new airport is shown on the coordinate plane.
Two additional runways are planned. Each will pass through the point (0, -4) 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.
(2 points)
Responses
y=−43x−4
y is equal to negative 4 thirds x minus 4
y=−34x−4
y is equal to negative 3 fourths x minus 4
y=34x−4
y is equal to 3 fourths x minus 4
y=43x−4
y is equal to 4 thirds x minus 4
Skip to navigation
The location of runway A for a new airport is shown on the coordinate plane.
Two additional runways are planned. Each will pass through the point (0, -4) 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.
(2 points)
Responses
y=−43x−4
y is equal to negative 4 thirds x minus 4
y=−34x−4
y is equal to negative 3 fourths x minus 4
y=34x−4
y is equal to 3 fourths x minus 4
y=43x−4
y is equal to 4 thirds x minus 4
Skip to navigation
Answers
GPT-4o mini
To determine the equations that could describe runways B and C, we need to know the slope of runway A. However, since the slope of runway A is not provided in the question, I'll demonstrate how to find parallel and perpendicular lines based on the general concept.
1. **Parallel Line (Runway B)**: A line that is parallel to another has the same slope. If the slope of runway A is \(m\), then the slope of runway B will also be \(m\). The equation of Runway B can be expressed in point-slope form as:
\[
y + 4 = m(x - 0) \implies y = mx - 4
\]
2. **Perpendicular Line (Runway C)**: A line that is perpendicular to another line has a slope that is the negative reciprocal of the original slope. If the slope of runway A is \(m\), then the slope of runway C will be \(-\frac{1}{m}\). The equation of Runway C can be expressed similarly:
\[
y + 4 = -\frac{1}{m}(x - 0) \implies y = -\frac{1}{m} x - 4
\]
### Analysis of Given Equations
Now let's analyze the given response options in terms of potential slopes that fit the description of parallel or perpendicular lines based on a hypothetical slope \(m\) for runway A.
- \(y = -\frac{4}{3}x - 4\): This line has a slope of \(-\frac{4}{3}\).
- \(y = -\frac{3}{4}x - 4\): This line has a slope of \(-\frac{3}{4}\).
- \(y = \frac{3}{4}x - 4\): This line has a slope of \(\frac{3}{4}\).
- \(y = \frac{4}{3}x - 4\): This line has a slope of \(\frac{4}{3}\).
### Conclusion
Without knowing the slope of runway A, we theoretically cannot assert which of the equations represent the runways. However, if we assume a slope for runway A:
- If runway A has a slope of 1 (for example), runways B and C could logically have slopes (1) and (-1) respectively.
- If the slope is \(-\frac{4}{3}\), then B would be parallel \(-\frac{4}{3}\) and C would be \(\frac{3}{4}\).
In the latter example, this means among the given equations, only **the options with \(-\frac{4}{3}\) and \(\frac{3}{4}\)** could justify.
### Answer
- \(y = -\frac{4}{3}x - 4\) (could be Runway B if A's slope is \(-\frac{4}{3}\))
- \(y = \frac{3}{4}x - 4\) (could be Runway C if A's slope is \(-\frac{4}{3}\))
However, you must know the specific slope of runway A to give a definitive answer.
1. **Parallel Line (Runway B)**: A line that is parallel to another has the same slope. If the slope of runway A is \(m\), then the slope of runway B will also be \(m\). The equation of Runway B can be expressed in point-slope form as:
\[
y + 4 = m(x - 0) \implies y = mx - 4
\]
2. **Perpendicular Line (Runway C)**: A line that is perpendicular to another line has a slope that is the negative reciprocal of the original slope. If the slope of runway A is \(m\), then the slope of runway C will be \(-\frac{1}{m}\). The equation of Runway C can be expressed similarly:
\[
y + 4 = -\frac{1}{m}(x - 0) \implies y = -\frac{1}{m} x - 4
\]
### Analysis of Given Equations
Now let's analyze the given response options in terms of potential slopes that fit the description of parallel or perpendicular lines based on a hypothetical slope \(m\) for runway A.
- \(y = -\frac{4}{3}x - 4\): This line has a slope of \(-\frac{4}{3}\).
- \(y = -\frac{3}{4}x - 4\): This line has a slope of \(-\frac{3}{4}\).
- \(y = \frac{3}{4}x - 4\): This line has a slope of \(\frac{3}{4}\).
- \(y = \frac{4}{3}x - 4\): This line has a slope of \(\frac{4}{3}\).
### Conclusion
Without knowing the slope of runway A, we theoretically cannot assert which of the equations represent the runways. However, if we assume a slope for runway A:
- If runway A has a slope of 1 (for example), runways B and C could logically have slopes (1) and (-1) respectively.
- If the slope is \(-\frac{4}{3}\), then B would be parallel \(-\frac{4}{3}\) and C would be \(\frac{3}{4}\).
In the latter example, this means among the given equations, only **the options with \(-\frac{4}{3}\) and \(\frac{3}{4}\)** could justify.
### Answer
- \(y = -\frac{4}{3}x - 4\) (could be Runway B if A's slope is \(-\frac{4}{3}\))
- \(y = \frac{3}{4}x - 4\) (could be Runway C if A's slope is \(-\frac{4}{3}\))
However, you must know the specific slope of runway A to give a definitive answer.