Question

Jessica 🤍 is making a map for school and coordinate plane. To hallways in the school perpendicular to each other. The main hallway passes through the points (-five, nine) and (four, six) the perpendicular hallway passes through the point (-4, three) equations for the lines that represent main hallway and the perpendicular hallway

Answer 1

To find the equations of the main hallway and the perpendicular hallway, we can follow these steps:

### Step 1: Find the equation of the main hallway

**Given points on the main hallway:**
Point A \((-5, 9)\)
Point B \((4, 6)\)

**Step 1.1: Calculate the slope of the main hallway.**

The slope \(m\) is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting in the coordinates of points A and B:
\[
m = \frac{6 - 9}{4 - (-5)} = \frac{-3}{4 + 5} = \frac{-3}{9} = -\frac{1}{3}
\]

**Step 1.2: Find the equation of the line using point-slope form.**

The point-slope form of a line is:
\[
y - y_1 = m(x - x_1)
\]
Using point A \((-5, 9)\):
\[
y - 9 = -\frac{1}{3}(x - (-5))
\]
\[
y - 9 = -\frac{1}{3}(x + 5)
\]
Distributing the slope:
\[
y - 9 = -\frac{1}{3}x - \frac{5}{3}
\]
Adding 9 to both sides to isolate y:
\[
y = -\frac{1}{3}x - \frac{5}{3} + 9
\]
Converting 9 to a fraction:
\[
9 = \frac{27}{3}
\]
So we have:
\[
y = -\frac{1}{3}x + \frac{27 - 5}{3}
\]
\[
y = -\frac{1}{3}x + \frac{22}{3}
\]

**Equation of the main hallway:**
\[
y = -\frac{1}{3}x + \frac{22}{3}
\]

### Step 2: Find the equation of the perpendicular hallway

**Given point on the perpendicular hallway:**
Point C \((-4, 3)\)

**Step 2.1: Calculate the slope of the perpendicular hallway.**

The slope of a line perpendicular to another is the negative reciprocal of the original slope. Since the slope of the main hallway is \(-\frac{1}{3}\), the slope of the perpendicular hallway \(m_{\perpendicular}\) is:
\[
m_{\perpendicular} = -\frac{1}{-\frac{1}{3}} = 3
\]

**Step 2.2: Find the equation of the perpendicular line using point-slope form.**

Using point C \((-4, 3)\):
\[
y - 3 = 3(x - (-4))
\]
\[
y - 3 = 3(x + 4)
\]
Distributing the slope:
\[
y - 3 = 3x + 12
\]
Adding 3 to both sides:
\[
y = 3x + 15
\]

**Equation of the perpendicular hallway:**
\[
y = 3x + 15
\]

### Summary of equations:

- **Main hallway:** \(y = -\frac{1}{3}x + \frac{22}{3}\)
- **Perpendicular hallway:** \(y = 3x + 15\)