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: \(y = 3x + 15\)
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