What is the Pythagorean Theorem?

The Pythagorean Theorem is a relation in a right-angled triangle. The rule states that #a^2 + b^2 = c^2# , in which #a# and #b# are the opposite and the adjacent sides, the 2 sides which make the right-angle, and #c# representing the hypotenuse, the longest side of the triangle. So if you have #a = 6# and #b = 8#, #c# would equal to #(6^2 + 8^2)^(1/2)#, (#x^(1/2)# meaning square rooted), which is equal to 10, #c#, the hypotenuse.

The Pythagorean Thereom (found by Pythagoras aka Pythagoras of Samos) is used to find the length of a side of a right triangle using the formula #a^2+b^2=c^2#!

A right triangle has two "legs" and a hypotenuse. A hypotenuse is the longest side of a right triangle and is always the opposite of the right angle corner. The legs can be a or b (it doesn't matter which is #a# or which is #b#). The #c# is always longer than #a# and #b#! To get some more clarity, take a look at the example down below!

In this case, lets say that #a# is #3#, #b# is #4# and #c# is #x#.

#a^2+b^2=c^2#
After substituting...
#3^2+4^2=x^2#
After simplifying...
#9+16=x^2#
Now, solve it!
#x^2=25#
Whoa, whoa, wait a second before you finalize that as the answer! We can simplify this. It's just not #x#, it's #x^2#! So we have to find the square root of #25# so that you can get your final answer! The square root of #25# is #5#. So...
#x=5#!

Remember, we don't use the Pythagorean Theorem just for the hypotenuse! We can use it for the other sides, too! Ex:

In this problem, we know the hypotenuse, but we need to find out what one of the "legs" is. Lets say that #6# is #a#, #x# is #b# and we know that #10# has to be #c#.

#a^2+b^2=c^2#
After substituting...
#6^2+x^2=10^2#
After simplifying...
#36+x^2=100#
Leave #x^2# on one side...
#x^2=100-36#
#x^2=64#
#x=8#

There! We have it! I hope you have a better clarity of the Pythagorean Thereom and understand it! My source (despite the images) is my mind! Sorry if my answer is too long!