Question

Find the length of the side of a square with diagonal of length $12$.

Short answer

The length of side is $6\sqrt{2}$.

Step-by-step solution

Step 1. Given information.

The square has a diagonal of length $12$.

Step 2. Concept used.

1. A square has all four sides are equal and each interior angle is right angle.
2. 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. Calculation.

First we draw a square with diagonal $12$and labeled $\square ABCD$.

Let side be x.

In $\Delta ADC$.

$\angle ADC={90}^{°}\text{and}\overrightarrow{AD}=\overrightarrow{DC}=x$

Therefore,$\Delta ADC$ is a right angle triangle.

We use Pythagoras Theorem.

$\begin{array}{c}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}\mathbf{=}{\mathbf{12}}^{\mathbf{2}}\\ \mathbf{2}{\mathbf{x}}^{\mathbf{2}}\mathbf{=}\mathbf{144}\\ {\mathbf{x}}^{\mathbf{2}}\mathbf{=}\mathbf{72}\end{array}$

Taking square root we get.

$\begin{array}{l}x=\pm \sqrt{72}\\ x=\pm 6\sqrt{2}\end{array}$

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

Therefore, the length of side is $6\sqrt{2}$.