Question

The location of runway A for a new airport is shown on the coordinate plane below.
the main points are (-4,0) (0,3)



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 find the equations of runways B and C in relation to runway A, we first need to determine the slope of runway A using the given points \((-4, 0)\) and \((0, 3)\).

1. **Calculating the slope of runway A:**
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 0}{0 - (-4)} = \frac{3}{4}
\]

The slope of runway A is \(\frac{3}{4}\).

2. **Finding the equation for runway B (parallel to A):**
The slope of runway B will be the same as runway A, which is \(\frac{3}{4}\). We can use the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) = (0, -4) \) and \( m = \frac{3}{4} \).

Plugging in these values:
\[
y - (-4) = \frac{3}{4}(x - 0)
\]
\[
y + 4 = \frac{3}{4}x
\]
\[
y = \frac{3}{4}x - 4
\]

Therefore, the equation for runway B is:
\[
y = \frac{3}{4}x - 4
\]

3. **Finding the equation for runway C (perpendicular to A):**
The slope of a line perpendicular to runway A is the negative reciprocal of \(\frac{3}{4}\). Thus, the slope for runway C is:
\[
m = -\frac{4}{3}
\]

Again using point-slope form with the point \((0, -4)\):
\[
y - (-4) = -\frac{4}{3}(x - 0)
\]
\[
y + 4 = -\frac{4}{3}x
\]
\[
y = -\frac{4}{3}x - 4
\]

Therefore, the equation for runway C is:
\[
y = -\frac{4}{3}x - 4
\]

Now, to summarize the equations we found:
- Runway B: \(y = \frac{3}{4}x - 4\)
- Runway C: \(y = -\frac{4}{3}x - 4\)

**Now, selecting the correct choices from the given options:**
- **A:** \(y = -\frac{3}{4}x - 4\) (Not correct)
- **B:** \(y = \frac{4}{3}x - 4\) (Not correct)
- **C:** \(y = -\frac{4}{3}x - 4\) (Correct, corresponds to Runway C)
- **D:** \(y = \frac{3}{4}x - 4\) (Correct, corresponds to Runway B)

Thus, the correct options are:
- **C**: \(y = -\frac{4}{3}x - 4\)
- **D**: \(y = \frac{3}{4}x - 4\)