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.