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.
Prove that the shortest distance between a point P, and a line, l, is the perpendicular line from P to l.
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 PB2= PA2+ AB2. Since both PA and AB have non-zero length, their squares are positive numbers.
If PB2 is equal to PA2 plus some positive number (AB2), it is bigger than PA2 and thus PB > PA, for every point B other than the one where the perpendicular line meets the line l.
(1) PA ⊥l //Given
(2) ΔPAB is a right triangle //Definition of a right triangle, from (1)
(3) PB2= PA2+ AB2 //Pythagorean Theorem
(4) AB2>0 //Square of a non-zero number is positive
(5) PB2>PA2 //(3),(4)
(6) PB>PA //Lines have positive lengths, so the square root is positive.