To determine which equations could describe runways B and C based on the location of runway A, we need to analyze the characteristics of each proposed runway.
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Runway A: Without the specific equation for runway A given, let's say its slope is \( m \). The slope decision will guide the slopes for the parallel and perpendicular lines.
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Runway B: This runway is parallel to runway A. Thus, it will have the same slope as runway A.
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Runway C: This runway is perpendicular to runway A. The slope of perpendicular lines is the negative reciprocal of the slope of the original line. If the slope of runway A is \( m \), the slope of runway C will be \( -\frac{1}{m} \).
Considering the point (0, -4) through which both runways B and C pass, we can determine their equations.
We need to identify the slopes from the given options:
- A. \( y = -\frac{3}{4}x - 4 \) has a slope of \( -\frac{3}{4} \).
- B. \( y = \frac{4}{3}x - 4 \) has a slope of \( \frac{4}{3} \).
- C. \( y = -\frac{4}{3}x - 4 \) has a slope of \( -\frac{4}{3} \).
- D. \( y = \frac{3}{4}x - 4 \) has a slope of \( \frac{3}{4} \).
Now we analyze which slopes can be derived from each other (for parallelism and perpendicularity):
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If Runway A had a slope of \( -\frac{3}{4} \) (as in option A), then:
- Runway B (parallel) would also have a slope \( -\frac{3}{4} \).
- Runway C (perpendicular) would then have a slope \( \frac{4}{3} \) (which is option B).
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If Runway A had a slope of \( \frac{4}{3} \) (as in option B), then:
- Runway B (parallel) would also have a slope of \( \frac{4}{3} \).
- Runway C (perpendicular) would then have a slope of \( -\frac{3}{4} \) (which is option D).
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If Runway A had a slope of \( -\frac{4}{3} \) (as in option C), then:
- Runway B (parallel) would also have a slope of \( -\frac{4}{3} \).
- Runway C (perpendicular) would then have a slope of \( \frac{3}{4} \) (which is option D).
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If Runway A had a slope of \( \frac{3}{4} \) (as in option D), then:
- Runway B (parallel) would also have a slope of \( \frac{3}{4} \).
- Runway C (perpendicular) would then have a slope of \( -\frac{4}{3} \) (which is option C).
Possible equations based on the above reasoning would be:
So, options that satisfy the conditions could be:
- For Runway B: If runway A's slope corresponds to one of the options, its parallel line could be the same slope.
- For Runway C: If the slopes are the negatives of these values (negative reciprocal), then those will apply.
Option associations would be:
- A (= Runway A) indicates B is option A itself; and C is option B
- B indicates B is option B and C is option D.
- C indicates B is option C and C is option D.
- D indicates B is option D and C is option C.
Thus, the correct equations that could describe either of the runways are options:
- A (if parallel) and B (perpendicular)
- C (if parallel) and D (perpendicular)
So:
- B: \( y = \frac{4}{3}x - 4 \)
- C: \( y = -\frac{4}{3}x - 4 \)
- D: \( y = \frac{3}{4}x - 4 \)
Hence the equations that could describe runways B and C are A, B, C, and D.
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