This is a problem involving a right-angled triangle, where angle $C = 90^\circ$. The two sides $a = 9$ and $b = 12$ are given, and the goal is to find side $c$ (the hypotenuse).

We can use the Pythagorean theorem to solve this:

$c^2 = a^2 + b^2$

Substitute the given values:

$c^2 = 9^2 + 12^2$ $c^2 = 81 + 144$ $c^2 = 225$

Now, take the square root of both sides:

$c = \sqrt{225} = 15$

So, the length of $c$ is 15.

Therefore, the correct answer is $c = 15$.

Would you like more details or have any further questions? Here are some related questions to explore:

1. How does the Pythagorean theorem work in non-right-angled triangles?
2. What would change if angle $C$ were not 90°?
3. How would you calculate the area of a right-angled triangle?
4. What are some real-life applications of the Pythagorean theorem?
5. Can the Pythagorean theorem be applied to 3D geometry?

Tip: Always double-check the type of triangle you're working with, as the Pythagorean theorem only applies to right-angled triangles.