Determine whether the lines through each pair of points are parallel, perpendicular, or neither. \( (-2,-13) \) and \( (0,-5)(0,-7) \) and \( (4,-6) \) Parallel Perpendicular Neither

To determine the relationship between the lines through the given pairs of points, we first need to find the slopes of the lines formed by each pair of points. 1. **Finding the slope of the line through the points \((-2, -13)\) and \((0, -5)\)**: The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] For the points \((-2, -13)\) and \((0, -5)\): \[ m_1 = \frac{-5 - (-13)}{0 - (-2)} = \frac{-5 + 13}{0 + 2} = \frac{8}{2} = 4 \] 2. **Finding the slope of the line through the points \((0, -7)\) and \((4, -6)\)**: For the points \((0, -7)\) and \((4, -6)\): \[ m_2 = \frac{-6 - (-7)}{4 - 0} = \frac{-6 + 7}{4} = \frac{1}{4} \] Now we have the slopes: - Slope of the first line \(m_1 = 4\) - Slope of the second line \(m_2 = \frac{1}{4}\) 3. **Determining the relationship**: - Two lines are **parallel** if their slopes are equal. - Two lines are **perpendicular** if the product of their slopes is \(-1\). Calculating the product of the slopes: \[ m_1 \cdot m_2 = 4 \cdot \frac{1}{4} = 1 \] Since the product of the slopes is \(1\), the lines are neither parallel nor perpendicular. **Final Answer**: Neither