Question

The location of runway A for a new airport is shown on the coordinate plane below.



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= - 3/4 x - 4

B. y= 4/3 x - 4

C. y= - 4/3 x - 4

D. y= 3/4 x - 4

Answer 1

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.

1. **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.

2. **Runway B**: This runway is parallel to runway A. Thus, it will have the same slope as runway A.

3. **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):

- 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).

- 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).

- 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).

- 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:
1. A (= Runway A) indicates B is option A itself; and C is option B
2. B indicates B is option B and C is option D.
3. C indicates B is option C and C is option D.
4. 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.**