Question

find the value of $x$.

Short answer

The value of $x$ is $8$.

Step-by-step solution

Step 1. Given information.    

The two sides of triangle are same that is $10$ and third side is $12$.

Step 2. Concept used.

1. In isosceles triangle, the vertex angle bisector of isosceles triangle is perpendicular bisector of the base.
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. Now find the value.

$\begin{array}{c}BC=CD\left(byconcept1\right)\\ \\ BD=12\\ BC+CD=12\\ 2BC=12\\ BC=6\end{array}$

Now we can use Pythagoras theorem in $\Delta ABC$.

$\begin{array}{c}{x}^2+6^2={10}^2\\ x^2+36=100\\ x^2=64\end{array}$

Taking square root we get.

$\begin{array}{l}x=\pm \sqrt{64}\\ x=\pm 8\end{array}$

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

Therefore, the value of x is 8.