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Home » Lines and Angles » Perpendicular lines » Shortest Distance Theorem

Shortest Distance Theorem

Last updated: Mar 27, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

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.

shortest distance theorem

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 a 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

Here's how you prove the Shortest Distance Theorem:

(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.

« Finding the Area of an Equilateral Triangle from its Perimeter
Triangle Inequality Theorem »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

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    • Pythagorean Theorem
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    • Similar Triangles
    • Triangle Inequalities

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