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=-3/4x-4
B. y=4/3x-4
C. y=-4/3x-4
D. y=3/4x-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.
A. y=-3/4x-4
B. y=4/3x-4
C. y=-4/3x-4
D. y=3/4x-4
Answers
GPT-4o mini
To determine the equations for runways B and C in relation to runway A, we need to analyze the slope of runway A, which is required to find the parallel and perpendicular slopes.
**Step 1: Identify the slope of runway A.**
Since we don't have the specific equation of runway A provided, let's assume runway A has a certain slope \( m_A \). If we assume two points on runway A, the slope can be calculated from the differences in the y-values and x-values of those points.
**Step 2: Calculate the slope for runways B and C:**
- **Runway B (parallel to runway A):**
If runway B is parallel to runway A, it shares the same slope as runway A. Therefore, the equation for runway B can be written as:
\[
y = m_A x + b
\]
where \( b \) is determined by the y-intercept.
- **Runway C (perpendicular to runway A):**
The slope of runway C will be the negative reciprocal of the slope of runway A. If the slope of runway A is \( m_A \), the slope of runway C is:
\[
m_C = -\frac{1}{m_A}
\]
The equation for runway C can be expressed similarly:
\[
y = m_C x + b
\]
**Step 3: Evaluate the options:**
To assess which equations are aligned with either runway B or C based on the conditions mentioned:
1. All equations are given in the form \( y = mx + b \), so we can extract the slopes.
- **A. \( y = -\frac{3}{4}x - 4 \)** → slope = \(-\frac{3}{4}\)
- **B. \( y = \frac{4}{3}x - 4 \)** → slope = \(\frac{4}{3}\)
- **C. \( y = -\frac{4}{3}x - 4 \)** → slope = \(-\frac{4}{3}\)
- **D. \( y = \frac{3}{4}x - 4 \)** → slope = \(\frac{3}{4}\)
**Step 4: Find slopes and determining relationships:**
Assuming the slope of runway A (for example) is either \(-\frac{3}{4}\) or \(\frac{4}{3}\):
- If \( m_A = -\frac{3}{4} \):
- Runway B (parallel) → slope: \(-\frac{3}{4}\) (matches option A).
- Runway C (perpendicular) → slope: \(\frac{4}{3}\) (matches option B).
- If \( m_A = \frac{4}{3} \):
- Runway B (parallel) → slope: \(\frac{4}{3}\) (matches option B).
- Runway C (perpendicular) → slope: \(-\frac{3}{4}\) (matches option D).
Following through with the slopes:
Based on the information, the corresponding answers for **runway B and runway C** can be:
- **For B (parallel):** A or B
- **For C (perpendicular):** C or D
Thus, if one were to choose equations being provided:
- **A.** could represent runway B if A has slope \(-\frac{3}{4}\).
- **B.** could represent runway B if A has slope \(\frac{4}{3}\).
- **C.** could represent runway C if A has slope \(-\frac{4}{3}\).
- **D.** could represent runway C if A has slope \(\frac{3}{4}\).
### Conclusions
Thus the correct choices are:
**A, B, C, D**
**Step 1: Identify the slope of runway A.**
Since we don't have the specific equation of runway A provided, let's assume runway A has a certain slope \( m_A \). If we assume two points on runway A, the slope can be calculated from the differences in the y-values and x-values of those points.
**Step 2: Calculate the slope for runways B and C:**
- **Runway B (parallel to runway A):**
If runway B is parallel to runway A, it shares the same slope as runway A. Therefore, the equation for runway B can be written as:
\[
y = m_A x + b
\]
where \( b \) is determined by the y-intercept.
- **Runway C (perpendicular to runway A):**
The slope of runway C will be the negative reciprocal of the slope of runway A. If the slope of runway A is \( m_A \), the slope of runway C is:
\[
m_C = -\frac{1}{m_A}
\]
The equation for runway C can be expressed similarly:
\[
y = m_C x + b
\]
**Step 3: Evaluate the options:**
To assess which equations are aligned with either runway B or C based on the conditions mentioned:
1. All equations are given in the form \( y = mx + b \), so we can extract the slopes.
- **A. \( y = -\frac{3}{4}x - 4 \)** → slope = \(-\frac{3}{4}\)
- **B. \( y = \frac{4}{3}x - 4 \)** → slope = \(\frac{4}{3}\)
- **C. \( y = -\frac{4}{3}x - 4 \)** → slope = \(-\frac{4}{3}\)
- **D. \( y = \frac{3}{4}x - 4 \)** → slope = \(\frac{3}{4}\)
**Step 4: Find slopes and determining relationships:**
Assuming the slope of runway A (for example) is either \(-\frac{3}{4}\) or \(\frac{4}{3}\):
- If \( m_A = -\frac{3}{4} \):
- Runway B (parallel) → slope: \(-\frac{3}{4}\) (matches option A).
- Runway C (perpendicular) → slope: \(\frac{4}{3}\) (matches option B).
- If \( m_A = \frac{4}{3} \):
- Runway B (parallel) → slope: \(\frac{4}{3}\) (matches option B).
- Runway C (perpendicular) → slope: \(-\frac{3}{4}\) (matches option D).
Following through with the slopes:
Based on the information, the corresponding answers for **runway B and runway C** can be:
- **For B (parallel):** A or B
- **For C (perpendicular):** C or D
Thus, if one were to choose equations being provided:
- **A.** could represent runway B if A has slope \(-\frac{3}{4}\).
- **B.** could represent runway B if A has slope \(\frac{4}{3}\).
- **C.** could represent runway C if A has slope \(-\frac{4}{3}\).
- **D.** could represent runway C if A has slope \(\frac{3}{4}\).
### Conclusions
Thus the correct choices are:
**A, B, C, D**