Pythagorean theorem
The Pythagorean Theorem was named after famous Greek mathematician Pythagoras. It is an important formula that states the following: a^{2} + b^{2} = c^{2}
Looking at the figure above, did you make the following important observation? This may help us to see why the formula works.
 The red square has 2 triangles in it
 The blue square has also 2 triangles in it
 The black square has 4 of the same triangle in it
Therefore, area of red square + area of blue square = area of black square
Let a = the length of a side of the red square
Let b = the length of a side of the blue square
Let c = the length of a side of the black square
Therefore, a^{2} + b^{2} = c^{2}
Generally speaking, in any right triangle, let c be the length of the longest side (called hypotenuse) and let a and b be the length of the other two sides (called legs).
The theorem states that the length of the hypotenuse squared is equal to the length of side a squared plus the length of side b squared.
Written as an equation, c
^{2} = a
^{2} + b
^{2}
Thus, given two sides, the third side can be found using the formula.
We will illustrate with examples, but before proceeding, you should know how to find the square root of a number and how to solve equations using subtraction.
Examples showing how to use the Pythagorean theorem
Exercise #1
Let a = 3 and b = 4. Find c, or the longest side
c
^{2} = a
^{2} + b
^{2}
c
^{2} = 3
^{2} + 4
^{2}
c
^{2}= 9 + 16
c
^{2} = 25
c = √25
The sign (√) means square root
c = 5
Exercise #2
Let c = 10 and a = 8. Find b, or the other leg.
c
^{2} = a
^{2} + b
^{2}
10
^{2} = 8
^{2} + b
^{2}
100 = 64 + b
^{2}
100  64 = 64  64 + b
^{2} (minus 64 from both sides to isolate b
^{2} )
36 = 0 + b
^{2}
36 = b
^{2}
b = √36 = 6
Exercise #3
Let c = 13 and b = 5. Find a
c
^{2} = a
^{2}+ b
^{2}
13
^{2} = a
^{2} + 5
^{2}
169 = a
^{2} + 25
169  25 = a
^{2} + 2525
144 = a
^{2} + 0
144 = a
^{2}
a = √144 = 12
Take the Pythagorean theorem quiz below to see how well you understand this lesson.
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