Two additional runways are planned. Each will pass through the point (0,-4) on the coordinate plane.

Question

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

Answer 1

To determine the equations of runways A, B, and C, we need to analyze the conditions given:

1. Runway B is parallel to runway A: This means that the slope of runway B must be the same as the slope of runway A.
2. Runway C is perpendicular to runway A: This means that the slope of runway C must be the negative reciprocal of the slope of runway A.

Given that both runway B and runway C will pass through the point (0, -4), we can express the equations of the runways in slope-intercept form: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Since all the runways pass through the point (0, -4), the y-intercept \( b \) for all the equations will be \(-4\).

We need to find slopes that fit these descriptions. Let's calculate:

First, let's find the slopes from the given equations:

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 = \(\frac{3}{4}\)

Analyzing the options:

1. Runway A: Assume it has a slope \( m \). We can choose any slope from the options A to D.
2. Runway B (Parallel to A): Should have the same slope as A.
3. Runway C (Perpendicular to A): Should have a slope that is the negative reciprocal of the slope of A.

To find which options could represent runway A and consequently B and C, we can examine each one:

• If runway A has a slope of -\(\frac{3}{4}\) (Option A), then B would also have slope -\(\frac{3}{4}\) and C would have slope \(\frac{4}{3}\).

• If runway A has a slope of \(\frac{4}{3}\) (Option B), then B would also have slope \(\frac{4}{3}\) and C would have slope -\(\frac{3}{4}\).

• If runway A has a slope of -\(\frac{4}{3}\) (Option C), then B would have slope -\(\frac{4}{3}\) and C would have slope \(\frac{3}{4}\).

• If runway A has a slope of \(\frac{3}{4}\) (Option D), then B would have slope \(\frac{3}{4}\) and C would have slope -\(\frac{4}{3}\).

Conclusion:

Each option can serve as runway A. Therefore, we can identify the properties of B and C for each slope selection:

• If Option A:

  • A: y = -\(\frac{3}{4}x - 4\), B: y = -\(\frac{3}{4}x - 4\), C: y = \(\frac{4}{3}x - 4\)

• If Option B:

  • A: y = \(\frac{4}{3}x - 4\), B: y = \(\frac{4}{3}x - 4\), C: y = -\(\frac{3}{4}x - 4\)

• If Option C:

  • A: y = -\(\frac{4}{3}x - 4\), B: y = -\(\frac{4}{3}x - 4\), C: y = \(\frac{3}{4}x - 4\)

• If Option D:

  • A: y = \(\frac{3}{4}x - 4\), B: y = \(\frac{3}{4}x - 4\), C: y = -\(\frac{4}{3}x - 4\)

Therefore, the correct options representing either runway A, B, or C are:

A, B, C, and D.