Question

Find the value of $x$ from the given figure.

Short answer

The value of $x$ is $6$.

Step-by-step solution

Step 1. Given information.

All sides of a quadrilateral are given $x$ and one diagonal is $6\sqrt{2}$ as show in given figure.

Step 2. Concept used.

1. Opposite angles of equal sides in triangle are equal (isosceles triangle).
2. The sum of all angle of a triangle is\[{{180}^{\circ }}\].
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 value.

In $\Delta ABC$

$\begin{array}{c}\angle BAC=\angle BCA\left(byconcept\text{1}\right)\\ \angle BAC=\angle BCA={45}^{\circ }\\ \angle BAC+\angle BCA+\angle ABC={180}^{\circ }\left(byconcept2\right)\\ {45}^{\circ }+{45}^{\circ }+\angle ABC={180}^{\circ }\\ \angle ABC={90}^{\circ }\end{array}$

Therefore $\Delta ABC$is a right angle triangle.

Now we use Pythagoras Theorem in $\Delta ABC$:

$\begin{array}{c}{x}^2+x^2=6\sqrt{2}\\ 2x^2=36\times 2\\ x^2=36\end{array}$

Taking square root we get.

$\begin{array}{l}x=\pm \sqrt{36}\\ x=\pm 6\end{array}$

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

Therefore, the value of $x$is $6$.