Question
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.
A. y=-3/4x-4
B. y=4/3x-4
C. y=-4/3x-4
D. y=3/4x-4
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. y=-3/4x-4
B. y=4/3x-4
C. y=-4/3x-4
D. y=3/4x-4
Answers
Answered by
GPT-4o mini
To find the equations for runways B and C based on the specifications given about their relationships with runway A, we need to analyze the slopes.
1. **Finding the slope of runway A**:
- The phrase "runway B is to be parallel to runway A" indicates that it will have the same slope as runway A.
- The phrase "runway C is to be perpendicular to runway A" indicates that its slope will be the negative reciprocal of the slope of runway A.
2. **Analyzing the provided equations and finding their slopes**:
- **A. \( y = -\frac{3}{4}x - 4 \)** → Slope = \(-\frac{3}{4}\)
- **B. \( y = \frac{4}{3}x - 4 \)** → Slope = \(\frac{4}{3}\) (negative reciprocal of \(-\frac{3}{4}\))
- **C. \( y = -\frac{4}{3}x - 4 \)** → Slope = \(-\frac{4}{3}\)
- **D. \( y = \frac{3}{4}x - 4 \)** → Slope = \(\frac{3}{4}\) (negative reciprocal of \(-\frac{4}{3}\))
3. **Determine which equations fit the criteria for either runway B (parallel) or runway C (perpendicular)**:
- From equation A, if we take the slope \(-\frac{3}{4}\):
- For runway B (parallel), we need to keep the slope \(-\frac{3}{4}\). Thus, equation A is a possibility.
- For runway C (perpendicular), we would need a slope of \(\frac{4}{3}\), which corresponds to option B.
- If using equation A's slope \(-\frac{4}{3}\):
- For runway B (parallel), we would need a slope of \(-\frac{4}{3}\), which corresponds to option C.
- For runway C (perpendicular), the slope would be \(\frac{3}{4}\), which corresponds to option D.
4. **Summary of the slopes**:
- If runway A has a slope of \(-\frac{3}{4}\), then:
- Runway B can be represented by A (same slope: A).
- Runway C can be represented by B (perpendicular: B).
- If runway A has a slope of \(-\frac{4}{3}\), then:
- Runway B can be represented by C (same slope: C).
- Runway C can be represented by D (perpendicular: D).
5. **Conclusion**: Therefore, the equations that could describe either runway B or C are:
- **A.** \( y = -\frac{3}{4}x - 4 \) (can be runway A itself).
- **B.** \( y = \frac{4}{3}x - 4 \) (perpendicular to A if A was with slope \(-\frac{3}{4}\)).
- **C.** \( y = -\frac{4}{3}x - 4 \) (can serve as runway A with a slope of \(-\frac{4}{3}\)).
- **D.** \( y = \frac{3}{4}x - 4 \) (perpendicular to A if A had slope \(-\frac{4}{3}\)).
Thus, the correct choices are:
- **A**
- **B**
- **C**
- **D**
All options can describe either runway B or C depending on the slope of runway A being considered.
1. **Finding the slope of runway A**:
- The phrase "runway B is to be parallel to runway A" indicates that it will have the same slope as runway A.
- The phrase "runway C is to be perpendicular to runway A" indicates that its slope will be the negative reciprocal of the slope of runway A.
2. **Analyzing the provided equations and finding their slopes**:
- **A. \( y = -\frac{3}{4}x - 4 \)** → Slope = \(-\frac{3}{4}\)
- **B. \( y = \frac{4}{3}x - 4 \)** → Slope = \(\frac{4}{3}\) (negative reciprocal of \(-\frac{3}{4}\))
- **C. \( y = -\frac{4}{3}x - 4 \)** → Slope = \(-\frac{4}{3}\)
- **D. \( y = \frac{3}{4}x - 4 \)** → Slope = \(\frac{3}{4}\) (negative reciprocal of \(-\frac{4}{3}\))
3. **Determine which equations fit the criteria for either runway B (parallel) or runway C (perpendicular)**:
- From equation A, if we take the slope \(-\frac{3}{4}\):
- For runway B (parallel), we need to keep the slope \(-\frac{3}{4}\). Thus, equation A is a possibility.
- For runway C (perpendicular), we would need a slope of \(\frac{4}{3}\), which corresponds to option B.
- If using equation A's slope \(-\frac{4}{3}\):
- For runway B (parallel), we would need a slope of \(-\frac{4}{3}\), which corresponds to option C.
- For runway C (perpendicular), the slope would be \(\frac{3}{4}\), which corresponds to option D.
4. **Summary of the slopes**:
- If runway A has a slope of \(-\frac{3}{4}\), then:
- Runway B can be represented by A (same slope: A).
- Runway C can be represented by B (perpendicular: B).
- If runway A has a slope of \(-\frac{4}{3}\), then:
- Runway B can be represented by C (same slope: C).
- Runway C can be represented by D (perpendicular: D).
5. **Conclusion**: Therefore, the equations that could describe either runway B or C are:
- **A.** \( y = -\frac{3}{4}x - 4 \) (can be runway A itself).
- **B.** \( y = \frac{4}{3}x - 4 \) (perpendicular to A if A was with slope \(-\frac{3}{4}\)).
- **C.** \( y = -\frac{4}{3}x - 4 \) (can serve as runway A with a slope of \(-\frac{4}{3}\)).
- **D.** \( y = \frac{3}{4}x - 4 \) (perpendicular to A if A had slope \(-\frac{4}{3}\)).
Thus, the correct choices are:
- **A**
- **B**
- **C**
- **D**
All options can describe either runway B or C depending on the slope of runway A being considered.