Geometry - Perpendicular Lines And Planes

If A is a given point and CD a given line, then there is exactly one line running through A that is perpendicular to CD. If B is the point on line CD that also resides on the line running perpedicular to CD, then that line, AB, is the shortest distance from point A to line CD.

In a plane, if CD is a line and B a point on CD, then there is exactly one line through B perpendicular to CD. If B happens to be the midpoint of CD, then AB is called the perpendicular bisector of CD. Every point on AB is equidistant from C and D.

If a line QP is perpendicular to a plane at a point P, then it is perpendicular to every line in the plane which passes through P. This property is used by carpenters when they make sure that a door frame is perpendicular to the floor. Otherwise the door will rub on the floor, as someone who lives in an old house is likely to know.

A line will be perpendicular to a plane if it is perpendicular to two lines in the plane. The carpenter, in setting up the door frame, need not check every line with his or her square; two will do.

If perpendiculars are not confined to a single plane, there will be an infinitude of lines through B perpendicular to CD, all lying in the plane which is perpendicular to Figure 3. Illustration by Hans & Cassidy. Courtesy of Gale Group.
CD. If B is a midpoint, this plane will be the perpendicular-bisector plane of CD, and every point on this plane will be equidistant from C and D.

Two planes are perpendicular if one of the planes contains a line which is perpendicular to the other plane. The panels of folding screens, for example, stay perpendicular to the floor because the hinge lines are perpendicular to the floor.