Determine whether the lines through each pair of points are parallel, perpendicular, or neithe \[ \begin{array}{l}(9,4) \text { and }(3,5) \\ (-3,-3) \text { and }(3,-4) \\ \text { parallel } \\ \text { perpendicular } \\ \text { neither }\end{array} \] Need Help?

To determine the relationship between the lines through the given pairs of points, we need to find the slopes of each line. For the points (9, 4) and (3, 5), we calculate the slope (m₁) using the formula: \[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 4}{3 - 9} = \frac{1}{-6} = -\frac{1}{6} \] For the points (-3, -3) and (3, -4), we calculate the slope (m₂): \[ m_2 = \frac{-4 - (-3)}{3 - (-3)} = \frac{-4 + 3}{3 + 3} = \frac{-1}{6} \] Now let's analyze the relationship: - **Parallel lines** have equal slopes: \( m_1 = m_2 \) - **Perpendicular lines** have slopes that are negative reciprocals: \( m_1 \cdot m_2 = -1 \) In this case, \( m_1 = -\frac{1}{6} \) and \( m_2 = -\frac{1}{6} \). Since both slopes are equal: The lines through the points (9, 4) and (3, 5) and (−3, −3) and (3, −4) are **parallel**. First Line: The slopes of the first and second pairs of points are both -1/6, which means they run parallel to each other across the grid, never touching. Just like two super cool best friends who stick together at the mall! Second Line: On the flip side, if the slopes were negative reciprocals, like one being 1/2 and the other -2, we'd talk about good ol' perpendicular intersecting angles—think of them crisscrossing like street signs at a bustling intersection! Keep practicing those slope calculations for more fun!