³1username³2³2name³2text_default³2version³21³2ts³2980519068³2ip³2209.69.30.10³2data³2text³3The PythagoreanTheorem is attributed to the Greek philosopher and mathematician Pythagoras who lived in the 6th century B.C. The theorem states a relationship between the lengths of the sides of a right triangle. It says:


The sum of the squares of the length the sides of a right triangle is equal to the square 
of the length of the hypotenuse.


Visually, the theorem can be illustrated as follows:


Given a right triangle, with sides a and b and hypotenuse c, (Figure 1)


                                                     / l


                                                   /   l


                                      '''c'''         /      l 

    
                                              /        l 

   
                                           /           l  '''b'''

          
                                        /              l


                                     /________________ l


                                           '''a'''


                                      Figure 1


the hypotenuse is the side opposite the right (90 degree) angle in a right triangle.


Then, '''c'''^'''2''' = '''a'''^'''2''' + '''b'''^'''2''', or '''c''' = sqrt('''a'''^'''2''' + '''b'''^'''2''').


Certain sets of 3 integers are useful to remember as being Pythagorean triples, that is,
they are possible lengths of the sides of a right triangle. For example:


                     '''a'''                     '''b'''                       '''c'''


                     3                     4                       5


                     9                    12                      15



The PythagoreanTheorem is an important tool in the study of TrigonometricFunctions. 

----

Poser:  (3,4,5) is a pythagorean triplet since 3^2 + 4^2 = 5^2.  Which positive integers are not part of a pythagorean triplet?

³3minor³30³3newauthor³31³3summary³3*³2tscreate³2979797168³2keepts³2981947135³2id³21019³2host³2209.69.30.xxx³2revision³28³1username³2TormodMacDomhnall³2name³2text_default³2version³21³2ts³2981947135³2ip³2209.66.79.109³2data³2newauthor³31³3minor³30³3text³3The PythagoreanTheorem is attributed to the Greek philosopher and mathematician Pythagoras who lived in the 6th century B.C. The theorem states a relationship between the lengths of the sides of a right triangle. It says:


The sum of the squares of the length the sides of a right triangle is equal to the square 
of the length of the hypotenuse.


Visually, the theorem can be illustrated as follows:


Given a right triangle, with sides a and b and hypotenuse c, (Figure 1)


                                                     / l


                                                   /   l


                                      '''c'''         /      l 

    
                                              /        l 

   
                                           /           l  '''b'''

          
                                        /              l


                                     /________________ l


                                           '''a'''


                                      Figure 1


the hypotenuse is the side opposite the right (90 degree) angle in a right triangle.


Then, '''c'''^'''2''' = '''a'''^'''2''' + '''b'''^'''2''', or '''c''' = sqrt('''a'''^'''2''' + '''b'''^'''2''').


Certain sets of 3 integers are useful to remember as being Pythagorean triples, that is,
they are possible lengths of the sides of a right triangle. For example:


                     '''a'''                     '''b'''                       '''c'''


                     3                     4                       5


                     9                    12                      15



The PythagoreanTheorem is an important tool in the study of TrigonometricFunctions. 

----

Poser:  (3,4,5) is a pythagorean triplet since 3^2 + 4^2 = 5^2.  Which positive integers are not part of a pythagorean triplet?

----
         
Proof:  Draw right triangle with sides a,b,and c as above.  Turn an identical triangle 180 degrees and stick its c side to the original triangle's to form a rectangle of sides a and b.  Draw an identical rectangle perpendicular to the first with only a corner touching.  Draw squares of side a and b to connect the two rectangles into one large square of side (a+b).  From this diagram, the area of the large square is 
'''(a+b)'''^'''2''' = '''a'''^'''2''' + '''b'''^'''2''' + 4(.5ab) by the formulas for the areas of squares and triangles.  

Now form a square from four of the triangles by placing the a side of one triangle in line with the b side of another, so that all four sides of the figure are (a+b).  Note that within the large square is a square of side c.  From this diagram, the area of the large square is '''(a+b)'''^'''2''' = 4(.5ab) + '''c'''^'''2'''.

Since '''(a+b)'''^'''2'''=itself, '''a'''^'''2''' + '''b'''^'''2''' + 4(.5ab) = 4(.5ab) + '''c'''^'''2'''.

Therefore, '''a'''^'''2''' + '''b'''^'''2''' = '''c'''^'''2'''.
³3summary³3*proof added³2tscreate³2979797168³2keepts³2981988632³2id³21146³2host³2cobrand.bomis.com³2revision³29³1username³2RoseParks³2name³2text_default³2version³21³2ts³2981988632³2ip³2209.66.79.109³2data³2text³3The PythagoreanTheorem is attributed to the Greek philosopher and mathematician Pythagoras who lived in the 6th century B.C. The theorem states a relationship between the lengths of the sides of a right triangle. It says:


The sum of the squares of the length the sides of a right triangle is equal to the square 
of the length of the hypotenuse.


Visually, the theorem can be illustrated as follows:


Given a right triangle, with sides a and b and hypotenuse c, (Figure 1)


                                                     / l


                                                   /   l


                                      '''c'''         /      l 

    
                                              /        l 

   
                                           /           l  '''b'''

          
                                        /              l


                                     /________________ l


                                           '''a'''


                                      Figure 1


the hypotenuse is the side opposite the right (90 degree) angle in a right triangle.


Then, '''c'''^'''2''' = '''a'''^'''2''' + '''b'''^'''2''', or '''c''' = sqrt('''a'''^'''2''' + '''b'''^'''2''').


Certain sets of 3 integers are useful to remember as being Pythagorean triples, that is,
they are possible lengths of the sides of a right triangle. For example:


                     '''a'''                     '''b'''                       '''c'''


                     3                     4                       5


                     9                    12                      15



The PythagoreanTheorem is an important tool in the study of TrigonometricFunctions. 

----

Poser:  (3,4,5) is a pythagorean triplet since 3^2 + 4^2 = 5^2.  Which positive integers are not part of a pythagorean triplet?

----
         
Proof:  Draw right triangle with sides a,b,and c as above.  Turn an identical triangle 180 degrees and stick its c side to the original triangle's to form a rectangle of sides a and b.  Draw an identical rectangle perpendicular to the first with only a corner touching.  Draw squares of side a and b to connect the two rectangles into one large square of side (a+b).  From this diagram, the area of the large square is 
'''(a+b)'''^'''2''' = '''a'''^'''2''' + '''b'''^'''2''' + 4(.5ab) by the formulas for the areas of squares and triangles.  

Now form a square from four of the triangles by placing the a side of one triangle in line with the b side of another, so that all four sides of the figure are (a+b).  Note that within the large square is a square of side c.  From this diagram, the area of the large square is '''(a+b)'''^'''2''' = 4(.5ab) + '''c'''^'''2'''.

Since '''(a+b)'''^'''2'''=itself, '''a'''^'''2''' + '''b'''^'''2''' + 4(.5ab) = 4(.5ab) + '''c'''^'''2'''.

Therefore, '''a'''^'''2''' + '''b'''^'''2''' = '''c'''^'''2'''.----

I think a visual of the triangles involved in this proof would be very helpful- even if they are really crude like mine.
³3minor³30³3newauthor³31³3summary³3*Proof could use visuals.³2tscreate³2979797168³2keepts³2981994107³2id³21015³2host³2cobrand.bomis.com³2revision³210