Geometry/Chapter 4/Lesson 6
Contents
Introduction[edit]
 In this lesson, we will be reviewing the Pythagorean theorem. For more reading on this, see w:Pythagorean theorem.
Pythagorean theorem[edit]
The Pythagorean theorem is the worldwide famous geometric theorem that sets up the relationship between ^{2}, ^{2} and ^{2} in a right triangle. ^{2} represents the hyptoenuse, or the longest side opposite of a right angle in a righttriangle. The formula is as described:
^{2} + ^{2} = ^{2}
How do I use this theorem in a triangle problem?[edit]
First, it is important to note that the Pythagorean theorem has a few easy shortcuts to its geometric confusion. These are known as the Pythagorean triples. The triples are:
 , ,
 , ,
 , ,
 , ,
 , ,
The multiples of these numbers also work.
For example, let's say a triangle has the following numerical inputs:
 ^{2} = 18
 ^{2} = 24
 ^{2} = 30
...and with this problem, we are asked:
Do these lengths represent a right triangle? 

Answer: Yes, it is a Pythagorean triple of , and 
...but, let's changed this equation. What about ^{2} is ? Then this equation is not longer a Pythagorean triple, and therefore, we must plug in the numbers into the Pythagorean theorem equation:
 ^{2} + ^{2} = ^{2}
 ^{2} + ^{2} = ^{2}
 + =
 =
If is replaced with , then we know that we have changed the triangle being dealt with from a right triangle to an obtuse triangle. See the section 2.3 for more info.
How do I use this theorem in a "findx" problem?[edit]
How do we solve for "x" using the Pythagorean theorem? 

Answer: √ 
How do we solve for "x" using the Pythagorean theorem? 

Answer: 12 
How do we solve for "x" using the Pythagorean theorem? 


What is the converse of the Pythagorean Theorem?[edit]
Converse of the Pythagorean Theorem: If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
You can use the converse to determine if a triangle is acute, right or obtuse.
 Acute: ^{2} < ^{2} + ^{2}
 Obtuse: ^{2} > ^{2} + ^{2}
 Right: ^{2} = ^{2} + ^{2}
If we have the numbers , and , we automatically need to plug it into our Pythagorean Theorem equation.
 ^{2} = ^{2} + ^{2}
 = +
 =
 >
Now that we have worked out the problem: What triangle are we working with here? 

Answer: Obtuse triangle 
 Special Note
We can use the converse of the Pythagorean Theorem to check if the Pythagorean triples are right angles. For example, let us use the triple , and .
 ^{2} = ^{2} + ^{2}
 = +
 =
As you can see, this Pythagorean triple is, indeed, a right angle.
See also[edit]
Wikibooks has a book on the topic of Trigonometry/The_Pythagorean_Theorem. 