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 4 thirds x minus 4
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
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 4 thirds x minus 4
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
Answers
GPT-4o mini
To determine the correct equations for runways B and C based on being parallel and perpendicular to runway A, we need to consider the slope of runway A.
Let's denote the slope of runway A as \( m \).
1. **For Runway B**, which is parallel to runway A, it will have the same slope \( m \).
- The equation will have the format \( y = mx - 4 \) since it must pass through the point (0, -4).
2. **For Runway C**, which is perpendicular to runway A, it will have a slope that is the negative reciprocal of \( m \), which is \( -\frac{1}{m} \). The equation will have the form \( y = -\frac{1}{m}x - 4 \).
**Evaluating the given equations:**
- **Option 1**: \( y = \frac{4}{3}x - 4 \)
- This has a slope of \( \frac{4}{3} \). If runway A has this slope, then this is the equation for runway B (parallel).
- **Option 2**: \( y = -\frac{4}{3}x - 4 \)
- This has a slope of \( -\frac{4}{3} \), which is the negative reciprocal of \( \frac{3}{4} \). This would represent runway C (perpendicular to A if A has a slope of \( \frac{3}{4} \)).
- **Option 3**: \( y = -\frac{3}{4}x - 4 \)
- This has a slope of \( -\frac{3}{4} \). If runway A has a slope of \( \frac{4}{3} \), then this could not be perpendicular, but rather it would be a different line.
- **Option 4**: \( y = \frac{3}{4}x - 4 \)
- This has a slope of \( \frac{3}{4} \). If this is the slope of runway A, the line is in fact parallel to A, making this another candidate for runway B.
### Conclusions:
Based on the evaluation, the following equations could describe the runways:
1. \( y = \frac{4}{3}x - 4 \) (Runway B)
2. \( y = -\frac{4}{3}x - 4 \) (Runway C)
Therefore, the correct answers are:
- **Option 1**: \( y = \frac{4}{3}x - 4 \)
- **Option 2**: \( y = -\frac{4}{3}x - 4 \)
Options 3 and 4 are not correct in the context of defining either runway B or C relative to runway A.
Let's denote the slope of runway A as \( m \).
1. **For Runway B**, which is parallel to runway A, it will have the same slope \( m \).
- The equation will have the format \( y = mx - 4 \) since it must pass through the point (0, -4).
2. **For Runway C**, which is perpendicular to runway A, it will have a slope that is the negative reciprocal of \( m \), which is \( -\frac{1}{m} \). The equation will have the form \( y = -\frac{1}{m}x - 4 \).
**Evaluating the given equations:**
- **Option 1**: \( y = \frac{4}{3}x - 4 \)
- This has a slope of \( \frac{4}{3} \). If runway A has this slope, then this is the equation for runway B (parallel).
- **Option 2**: \( y = -\frac{4}{3}x - 4 \)
- This has a slope of \( -\frac{4}{3} \), which is the negative reciprocal of \( \frac{3}{4} \). This would represent runway C (perpendicular to A if A has a slope of \( \frac{3}{4} \)).
- **Option 3**: \( y = -\frac{3}{4}x - 4 \)
- This has a slope of \( -\frac{3}{4} \). If runway A has a slope of \( \frac{4}{3} \), then this could not be perpendicular, but rather it would be a different line.
- **Option 4**: \( y = \frac{3}{4}x - 4 \)
- This has a slope of \( \frac{3}{4} \). If this is the slope of runway A, the line is in fact parallel to A, making this another candidate for runway B.
### Conclusions:
Based on the evaluation, the following equations could describe the runways:
1. \( y = \frac{4}{3}x - 4 \) (Runway B)
2. \( y = -\frac{4}{3}x - 4 \) (Runway C)
Therefore, the correct answers are:
- **Option 1**: \( y = \frac{4}{3}x - 4 \)
- **Option 2**: \( y = -\frac{4}{3}x - 4 \)
Options 3 and 4 are not correct in the context of defining either runway B or C relative to runway A.