Question

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

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?

Answer 1

To find the equations of runways B and C based on the given runway A, we first need to determine the slope of runway A.

1. **Finding the slope of Runway A**:
- The endpoints of runway A are given as points \( (0, 3) \) and \( (-4, 0) \).
- We can calculate the slope \( m \) using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 3}{-4 - 0} = \frac{-3}{-4} = \frac{3}{4}
\]
- So, the slope of runway A is \( \frac{3}{4} \).

2. **Equation of Runway A**:
- Using the point-slope form of a line and point \( (0, 3) \):
\[
y - 3 = \frac{3}{4}(x - 0) \implies y = \frac{3}{4}x + 3
\]

3. **Equation of Runway B (parallel to Runway A)**:
- Runway B has the same slope as runway A, which is \( \frac{3}{4} \).
- To find the equation of runway B that passes through point \( (0, -4) \), we use the point-slope form:
\[
y - (-4) = \frac{3}{4}(x - 0) \implies y + 4 = \frac{3}{4}x \implies y = \frac{3}{4}x - 4
\]

4. **Equation of Runway C (perpendicular to Runway A)**:
- The slope of a line perpendicular to another is the negative reciprocal. Thus, the slope of runway C is:
\[
m = -\frac{1}{\frac{3}{4}} = -\frac{4}{3}
\]
- To find the equation of runway C passing through \( (0, -4) \):
\[
y - (-4) = -\frac{4}{3}(x - 0) \implies y + 4 = -\frac{4}{3}x \implies y = -\frac{4}{3}x - 4
\]

### Final equations:
- **Runway B**: \( y = \frac{3}{4}x - 4 \)
- **Runway C**: \( y = -\frac{4}{3}x - 4 \)

These are the equations for the two additional runways B and C as required.