Perpendicular Lines

Two nonvertical lines intersecting at a 90-degree angle are called perpendicular lines. Their slopes are negative reciprocals, so their product is negative one.

Imagine an engineer designing braces for a building. Two angled braces may need to meet at a right angle to keep the structure properly aligned. On a coordinate graph, these braces can be modeled as two nonvertical lines intersecting at 90 degrees.

We can prove the relationship between their slopes by placing the point of intersection at the origin. The equations of the lines can then be written as y equals m_1 x and y equals m_2 x.

Next, we choose a point on each line, labeled A and B. Because the lines are perpendicular, segments OA and OB form the legs of a right triangle, and AB is the hypotenuse. Applying the Pythagorean theorem and the distance formula gives an equation involving m_1 and m_2.

Simplifying this equation gives m_1 m_2 equals negative 1. So, two nonvertical lines are perpendicular if and only if the product of their slopes is negative one.

Engineers can use this relationship to check that angled braces meet at a right angle.