Over 2,500 years ago, a Greek mathematician named Pythagoras popularized the concept that a relationship exists between the hypotenuse and the legs of right triangles and that this relationship is true for all right triangles.  The Egyptians knew of this concept, as it related to 3, 4, 5 right triangles, long before the time of Pythagoras.  It was Pythagoras, however, who generalized the concept and who is attributed with its first geometrical demonstration.  Thus, it has become known as the Pythagorean Theorem.  


for right triangles

 

"In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs."


This relationship can be stated as:


for any right triangle

and is known as the
 Pythagorean Theorem.


a, b are legs.
   
c is the hypotenuse
(c is across from the right angle).

 

There are certain sets of numbers that have a very special property in relation to the Pythagorean Theorem.  Not only do these numbers satisfy the Pythagorean Theorem, but any multiples of these numbers also satisfy the Pythagorean Theorem.  

For example:  the numbers 3, 4, and 5 satisfy the Pythagorean Theorem.  If you multiply all three numbers by 2  (6, 8, and 10), these new numbers ALSO satisfy the Pythagorean theorem.  

  The special sets of numbers that possess this property are called 
Pythagorean Triples.

The most common Pythagorean Triples are:

3, 4, 5

5, 12, 13

8, 15, 17

 

REMEMBER: The Pythagorean Theorem ONLY works in Right Triangles!

 

 

Example 1:

Find x.

Answer:  10 m

This problem could also be solved using the Pythagorean Triple 3, 4, 5.  Since 6 is 2 times 3, and 8 is 2 times 4, then x must be 2 times 5.

 

Example 2:

A triangle has sides 6, 7 and 10. 
Is it a right triangle?

Let a = 6, b = 7 and c = 10.  The longest side MUST be the hypotenuse, so c = 10.  Now, check to see if the Pythagorean Theorem is true.

Since the Pythagorean Theorem is NOT true, this triangle is NOT a right triangle.

 

Example 3:

A ramp was constructed to load a truck.  If the ramp is 9 feet long and the horizontal distance from the bottom of the ramp to the truck is 7 feet, what is the vertical height of the ramp to the nearest tenth of a foot?

Since the ramp is described as having horizontal and vertical measurements, a right angle is implied.  Solve using the Pythagorean Theorem:

The height of the ramp is 5.7 feet.  The ramp will allow packages to be loaded into an area of the truck that is too high to be reached from the ground.