Distinct non-vertical parallel lines have equal gradients. Non-vertical perpendicular lines have gradient product -1: a rise/run vector (u,v) turns through a right angle to (-v,u). Find the reference gradient. Keep it for a parallel line, or use its negative reciprocal for a perpendicular line. Substitute the required point. A vertical line is perpendicular to a horizontal line. Perpendicular to y=2x+1 through (4,1): m=-1/2, so y-1=- 12(x-4), giving y=- 12x+3. Check: equal gradients can describe the same line, so check the intercept before calling lines distinct and parallel. Collinear points have the same connecting gradient, or all have the same x.

Distinct non-vertical parallel lines have equal gradients. Non-vertical perpendicular lines have gradient product \(-1\): a rise/run vector \((u,v)\) turns through a right angle to \((-v,u)\).

Find the reference gradient. Keep it for a parallel line, or use its negative reciprocal for a perpendicular line. Substitute the required point. A vertical line is perpendicular to a horizontal line.

Perpendicular to \(y=2x+1\) through \((4,1)\): \(m=-1/2\), so \(y-1=-\frac12(x-4)\), giving \(y=-\frac12x+3\).

Check: equal gradients can describe the same line, so check the intercept before calling lines distinct and parallel. Collinear points have the same connecting gradient, or all have the same \(x\).

Parallel and perpendicular constructions

Parallel and perpendicular constructions