# The diagonal of a square has length 12sqrt2 ft. How do you find the length of the side of the square?

Mar 9, 2018

length of side: $12$ feet

#### Explanation:

Since the figure is a square, it's sides have the same length; Lets call this length $s$

The diagonal forms a hypotenuse, $h$, with two of the sides
and, based on the Pythagorean Theorem,
$\textcolor{w h i t e}{\text{XXX}} {s}^{2} + {s}^{2} = {h}^{2}$
and since the diagonal ($h$) is given as having a length of $12 \sqrt{2}$,
we have
$\textcolor{w h i t e}{\text{XXX}} 2 {s}^{2} = {\left(12 \sqrt{2}\right)}^{2} = {12}^{2} \cdot {\left(\sqrt{2}\right)}^{2} = {12}^{2} \cdot 2$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow {s}^{2} = {12}^{2}$

and since (in the normal world) lengths can not be negative:
$\textcolor{w h i t e}{\text{XXX}} s = 12$

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An easier way to see this is to remember the standard right triangle (often used in trigonometry):

and realize the given triangle is a similar triangle simply scaled up by a factor of $12$