Slopes of Perpendicular Lines
- Ken Schwartz
- Difference and Slope
Why are perpendicular slopes negative reciprocals of each other?
A simple app to show how the slopes of two perpendicular lines relate to each other. You can drag the black intersection point around the graph, and once you find a spot for it, you can move the purple "Drag Me!" point to rotate the lines (changing the slope). The red and blue lines remain fixed as perpendiculars of each other. To measure the slope of each, the of the blue line and the of the red line are fixed at . By doing this, we can see that the blue and the red are of the same size but of opposite sign. Examining the graph with the lines at various angles should help solidify the concept that the slopes of perpendicular lines always have opposite signs: if one slopes up, the other has to slope down. Furthermore, the magnitudes of the rise and run of one equal the run and rise, respectively, of the other. Combining these two ideas gives us the relationship , where and are the slopes of two perpendicular lines. In the extreme case, if (horizontal), then , which is undefined, and is often represented as (vertical).