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Pythagorean Theorem

The Pythagorean Theorem states that
the sum of the squares
of the two small sides of a right triangle is equal to
the square of the third side.

This can be summarized by the equation
a^2 + b^2 = c^2
(The caret ("^")  is used here to indicate raised to a power.
a^2 = "a squared" = a x a)

The two short sides of a right triangle are
the sides that are connected to the right angle,
often indicated by a small box inside the corner.

When you know the lengths of the two short sides (a and b),
you can square each of them and add them together.
Then take the square root, and you have the length of
the third side (c)

For example, say that you know that the two short sides of a triangle
are 5 and 12.  Then let a = 5 and b = 12.

You must find the length of the
hypotenuse (the long side) which we will call c.

a^2 + b^2 = c^2
5^2 + 12^2 = c^2
25 + 144 = c^2
169 = c^2
so c = the square root of 169, which is 13 (13 x 13 = 169).

If you know the length of the hypotenuse and one other side, you can still
solve it using the Pythagorean Theorem.

For example, suppose you know that one of the short sides is 8 and the long
side is 10.  Then let a = 8 and c = 10.  Solve for b.

8^2 + b^2 = 10^2
64 + b^2 = 100

Subtract 64 from each side to get:
b^2 = 36
so b = 6.


Most problems that use the Pythagorean Theorem
utilize multiplies of what are known as Pythagorean Triples.

The first two are (3, 4, 5) and (5,12, 13).

While there are many more,
if you are aware of these triples and are
on the lookout for multiples of them

such as (6, 8, 10 or  10, 24, 26),

you can often find missing sides
without actually using the Pythagorean Theorem.

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