The shortest distance theorem states that the shortest distance between a point P, and a line, l, is the perpendicular line from P to l. It is also called the “perpendicular distance.”

It is simple to prove this theorem using the Pythagorean Theorem.

## Problem

Prove that the shortest distance between a point P, and a line, l, is the perpendicular line from P to l.

## Strategy

Since the perpendicular line from P to l forms a right angle, we will try to use what we know about right triangles, and the theorem we have about lengths of the sides of right triangle – the Pythagorean Theorem.

Any line from P to l other than the perpendicular line PA will form a right triangle, where that new line is the hypotenuse.

From the Pythagorean Theorem, we know that |PB|^{2}= |PA|^{2}+ |AB|^{2}. Since both PA and AB have non-zero length, their squares are positive numbers.

If |PB|^{2} is equal to |PA|^{2} plus some positive number (|AB|^{2}), it is bigger than |PA|^{2} and thus |PB| > |PA|, for every point B other than the one where the perpendicular line meets the line l.

## Proof

(1) PA ⊥l //Given

(2) ΔPAB is a right triangle //Definition of a right triangle, from (1)

(3) |PB|^{2}= |PA|^{2}+ |AB|^{2} //Pythagorean Theorem

(4) |AB|^{2}>0 //Square of a non-zero number is positive

(5) |PB|^{2}>|PA|^{2} //(3),(4)

(6) |PB|>|PA| //Lines have positive lengths, so the square root is positive.