What is the value of the $X$? 
With the details, what could be the value of X?

Let $d(x, y)$ be the distance of two vectors $x, y \in \mathbb{R}^2$. Let $p$ be the point where the two triangles touch. We aim to find $||p||$. $d((15/2, 0), p) = 15/2$ and $d((0, 10), p) = 5$. Let $p = (p_1, p_2)$. Then 
\begin{align*}
\sqrt{(15/2 - p_1)^2 + p_2^2} = 15/2, \ \sqrt{p_1^2 + (10-p_2)^2} = 5
\end{align*}
The first gives $p_2 = \sqrt{15 p_1 - p_1^2}$, assuming it is positive. Inserting this into the second we get 
\begin{align*}
\sqrt{p_1^2 + (10-\sqrt{15 p_1 - p_1^2})^2} = 5
\end{align*}
And solving for $p_1$ we get $p_1 = 3$, so that $p_2$ must be $6$, using the other equations.
$||p||$ must therefore be $\sqrt{ 3^2 + 6^2} = \sqrt{45} = 3 \sqrt{5}$
More detail upon request.

Edit: it has been pointed out by Andrei in the comments below that the equations that solve for the coordinates of $p$ can be geometrically interpreted as finding the point at the intersection of two circles. 
