This problem is based on Similar Triangles. [Refer attached Image]

Now, two triangles are similar when all of their angles are equal.

Let's find out all angles of these two triangles. [Refer attached Image]
Now, we know that all angles of these two triangles are equal, that means they are similar triangles
=> Triangle ACB ~ Triangle ECD and we know that
"If two triangles are similar then their corresponding sides are in the same ratio"
=> $\frac{AC}{EC}$ = $\frac{BC}{CD}$
Let's find out AC using Pythagoras theorem
$AB^2$ = $AC^2$ + $BC^2$
$10^2$ = $AC^2$ + $5^2$
$AC^2$ = 100 - 25 = 75
AC = $\sqrt{75}$ = 5$\sqrt{3}$

Using $\frac{AC}{EC}$ = $\frac{BC}{CD}$
CD = $\frac{BC*EC}{AC}$ = $\frac{5*10}{5\sqrt{3}}$
CD = $\frac{10}{\sqrt{3}}$

So, Answer is C
Hope it helps!