Perpendicular Slope Relation

Turn a line through a right angle and its rise and run swap places, with one of them changing sign — so a slope of 4 becomes −0.25, and 2/3 becomes −3/2. Equivalently m₁m₂ = −1, which is the version to memorise because it is symmetric and easy to check. Parallel lines are the trivial companion case: they simply share a slope, m₁ = m₂, and differ only in intercept.

The relation falls straight out of the Pythagorean theorem applied to the triangle formed by the two lines, and it is what lets coordinate geometry prove things classical geometry needed constructions for — that the diagonals of a rhombus meet at right angles, say, or that the altitudes of a triangle are concurrent. In practice it is how you write the equation of a perpendicular bisector: find the midpoint, take the negative reciprocal of the segment's slope, and drop both into point-slope form. The trap is doing only half the job. The perpendicular to y = 3x + 1 is not y = −3x + 1 (that is a reflection) and not y = x/3 + 1 (that is a reciprocal without the flip) but y = −x/3 + 1. Horizontal and vertical lines are the one exception the algebra cannot express: they are perpendicular to each other, but one has slope 0 and the other has none at all.