A circle has a chord of length $10$, and the distance from the center of the circle to the chord is $5$.  What is the area of the circle?

$\textbf{(A) }25\pi\qquad\textbf{(B) }50\pi\qquad\textbf{(C) }75\pi\qquad\textbf{(D) }100\pi\qquad\textbf{(E) }125\pi$
1. Let's denote the radius of the circle as \( r \). We are given that the chord has a length of \( 10 \) and the distance from the center of the circle to the chord is \( 5 \).

2. We can draw a perpendicular from the center of the circle to the chord, which will bisect the chord into two equal segments of length \( 5 \) each (since the total length of the chord is \( 10 \)).

3. This forms a right triangle where:
   - The hypotenuse is the radius \( r \) of the circle.
   - One leg is the distance from the center to the chord, which is \( 5 \).
   - The other leg is half the length of the chord, which is also \( 5 \).

4. Using the Pythagorean theorem for this right triangle, we have:
   \[
   r^2 = 5^2 + 5^2
   \]
   \[
   r^2 = 25 + 25
   \]
   \[
   r^2 = 50
   \]
   \[
   r = \sqrt{50}
   \]

5. The area \( A \) of the circle is given by the formula:
   \[
   A = \pi r^2
   \]
   Substituting \( r = \sqrt{50} \):
   \[
   A = \pi (\sqrt{50})^2
   \]
   \[
   A = \pi \cdot 50
   \]
   \[
   A = 50\pi
   \]

The final answer is \(\boxed{50\pi}\).