The Pythagorean Theorem was named after famous Greek mathematician Pythagoras. It is an important formula that states the following: a2 + b2 = c2
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, a2 + b2 = c2
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, c2
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
Let a = 3 and b = 4. Find c, or the longest side
= 9 + 16
c = √25
The sign (√) means square root
c = 5
Let c = 10 and a = 8. Find b, or the other leg.
100 = 64 + b2
100 - 64 = 64 - 64 + b2
(minus 64 from both sides to isolate b2
36 = 0 + b2
36 = b2
b = √36 = 6
Let c = 13 and b = 5. Find a
= a 2
169 = a2
169 - 25 = a2
144 = a2
144 = a2
a = √144 = 12
Take the Pythagorean theorem quiz below to see how well you understand this lesson.
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