Question

Find the length of the diagonal of the square with perimeter is $16$.

Short answer

The length of diagonal is $4\sqrt{2}$.

Step-by-step solution

Step 1. Given information.

Perimeter of a square is $16$.

Step 2. Concept used.

1. All four sides are equal of a square.
2. Perimeter of a square is $4a$, here a is side of square.
3. Pythagoras Theorem state that in a right angle triangle, the square of the hypotenuse is equal to the sum of squares of other sides.

Step 3. Now find the diagonal.

First we find out the side of square.

$\begin{array}{c}4a=16\\ a=4\end{array}$

Now we draw a square and labeled side and $\square ABCD$.

In $\Delta ADC$.

We use Pythagoras Theorem.

$\begin{array}{l}{x}^2=4^2+4^2\\ x^2=16+16\\ x^2=32\end{array}$

Taking square root we get.

$\begin{array}{l}x=\pm \sqrt{32}\\ x=\pm 4\sqrt{2}\end{array}$

Since, distance is always a positive quantity. So, we neglect negative sign.

Therefore, the diagonal of square is $4\sqrt{2}$.