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!