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# The diagonal of a square has length $12\sqrt{2}ft$ . How do you find the length of the side of the square?

Last updated date: 20th Jun 2024
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Hint: We are given a diagonal of a square has length $12\sqrt{2}ft$ . We are asked to find the side of the square. To do this, we start by first understanding what is square, what properties does it have then we use them to find the relation between the side and the diagonal, we will use that angle is ${{90}^{\circ }}$ , so we use Pythagoras theorem which say ${{a}^{2}}+{{b}^{2}}={{c}^{2}}$ using this we will find the length of the side.

We are given the diagonal of the square has length $12\sqrt{2}ft$ , we have to find the length of the side.
To do so we will understand the behavior of squares.
Square is a regular polygon with four sides as regular. It means it has all sides equal to each other and also all the angles are equal.
Each angle in the square is of ${{90}^{\circ }}$ .
Now diagonals are the line joining the opposite vertex of the square.

Here AC is diagonal, AB, BC, CD and DA are sides of the square.
Now as we know that in a right angle triangle we can use Pythagoras theorem which say that for right triangle ABC

$A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}}$
So as we know the angle in square is the right angle so from the figure we can see that ABC is a right angled triangle.
So, we will use Pythagoras theorem we have –
$A{{B}^{2}}+B{{C}^{2}}=A{{C}^{2}}$
Now, as we have AB and BC as side so they are equal. so, $AB=BC=S$ and $AC$ is diagonal and given as $12\sqrt{2}$
So, $AC=12\sqrt{2}$
Using these values, we get –
${{s}^{2}}+{{s}^{2}}={{\left( 12\sqrt{2} \right)}^{2}}$
Simplifying we get –
$2{{s}^{2}}=144\times 2=288$
Dividing both sides by ‘2’, we get –
${{s}^{2}}=144$
So as we know $144={{12}^{2}}$
So, ${{s}^{2}}={{12}^{2}}$
Hence, we get –
$s=12$

So, the length of the side is 12 ft.

Note: Remember that if the figure has all sides equal to each other than that figure is not necessary to be square. There is a figure called rhombus, in rhombus all sides are equal but they are not square. For the square side must be equal and angles should be ${{90}^{\circ }}$ each than only it is called as a square.