Shortest Distance
Two results follow from Theorems 1 and 2:
Corollary 1:
The shortest distance from a point to a line is the length of the perpendicular segment from that point to that line.
Proof:
Let be a point not on the line, let be the perpendicular segment from to the line, and suppose point is some other point on the line. Draw segment . Then is a right triangle with . Since the angles of a triangle add to 180o, the other two angles in this triangle must be less than 90o. Therefore, , so the opposite sides have the same inequality: . That is, any other segment from to the line is longer than .
Corollary 2:
The perpendicular segment from a point to a plane is the shortest distance from that point to the plane.
Proof:
Let be a point not in the plane, let be the perpendicular segment from to the plane, and suppose point is some other point in the plane. Then is shorter than because it is the shortest segment to line , by Corollary 1.