Perpendicular lines : four proofs of the negative reciprocal relationship

Abstract

One of the great joys of mathematics is finding multiple ways of arriving at a solution or proving a result. Some approaches can be messy and longwinded, while others can be elegant and succinct. But the joy lies in arriving at a common endpoint, however circuitous the route may have been. In this article we explore four different ways of proving the negative reciprocal relationship between the gradients of perpendicular lines. Each proof uses elementary ideas from other topics encountered in the high school Mathematics curriculum.