Section 84
The Pythagorean Theorem
The
Pythagorean Theorem
is named after
Pythagoras of Samos
, a mathematician who was also a religious leader, and believed that all things in the universe were composed of numbers.
He is supposed to have been the first to have proved this theorem about right triangles:
Pythagorean Theorem.
In a right triangle with legs of lengths
a
and
b
and
hypotenuse
of length
c
, the following equation is true:
c
^{
2
}
=
a
^{
2
}
+
b
^{
2
}
(There are many different ways to prove this.)
Below is a graphical representation. The theorem states that the sum of the areas of the blue and red squares is equal to the area of the green square.
Important
: Remember that the Pythagorean Theorem is true
only
for right triangles – triangles which have a 90
^{
o
}
angle.
The converse of the theorem is also true: if a triangle has sides of lengths
a
,
b
, and
c
, and
c
^{
2
}
=
a
^{
2
}
+
b
^{
2
}
, then it must be a
right
triangle
.
PYTHAGOREAN TRIPLES
Three whole numbers
a
,
b
,
c
which satisfy the equation of the Pythagorean Theorem are called
Pythagorean triples
. A few of the smallest ones are shown in the table below. Each Pythagorean Triple corresponds with a right triangle whose side lengths are in wholenumber ratios.
Pythagorean Triples

3, 4, 5

3
^{
2
}
+ 4
^{
2
}
= 5
^{
2
}
9 + 16 = 25

6, 8, 10

6
^{
2
}
+ 8
^{
2
}
= 10
^{
2
}
36 + 64 = 100

5, 12, 13

5
^{
2
}
+ 12
^{
2
}
= 13
^{
2
}
25 + 144 = 169

8, 15, 17

8
^{
2
}
+ 15
^{
2
}
= 17
^{
2
}
64 + 225 = 289
