# What is the Pythagorean Theorem?

Nov 24, 2014

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 ${\left({6}^{2} + {8}^{2}\right)}^{\frac{1}{2}}$, (${x}^{\frac{1}{2}}$ meaning square rooted), which is equal to 10, $c$, the hypotenuse. Aug 24, 2017

#### Explanation:

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!
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