Parallel and Perpendicular Lines

Objective: Find the slope of a line parallel to a given line.

Prerequisite skills: slope and line equations. Distinct parallel lines have the same direction and never intersect in the plane. For nonvertical lines this means equal slopes and different intercepts. Equal slopes with equal intercepts describe the same line. Common error: taking a negative reciprocal for parallel lines. Restrictions and interpretation: Parallel distinct nonvertical lines have equal slopes; horizontal and vertical lines form a special perpendicular pair.

Objective: Use a negative reciprocal to find a perpendicular direction.

When two nonvertical, nonhorizontal lines are perpendicular, their slopes multiply to −1. This exchanges rise and run and reverses one direction. A horizontal line is perpendicular to a vertical line, whose slope is undefined. Common error: changing the sign without taking a reciprocal. Restrictions and interpretation: Parallel distinct nonvertical lines have equal slopes; horizontal and vertical lines form a special perpendicular pair.

Objective: Combine a direction condition with a known point.

Find the required slope first, then use the known point to determine the intercept. Meeting the slope condition alone gives a family of lines, so the point is essential. Common error: using the old line intercept in a new line. Restrictions and interpretation: Parallel distinct nonvertical lines have equal slopes; horizontal and vertical lines form a special perpendicular pair. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.