To find the length of the diagonal of rectangle $PQRS$, we use the Pythagorean theorem, which states:

$\text{Diagonal}^2 = \text{Length}^2 + \text{Width}^2$

Step-by-step solution:

1. The given length of the rectangle is $12 \, \text{cm}$, and the width is $5 \, \text{cm}$.
2. Substitute these values into the Pythagorean theorem:

$\text{Diagonal}^2 = 12^2 + 5^2$

$\text{Diagonal}^2 = 144 + 25$

$\text{Diagonal}^2 = 169$

3. Take the square root to find the diagonal:

$\text{Diagonal} = \sqrt{169} = 13 \, \text{cm}$

Final Answer:

The length of the diagonal is $13 \, \text{cm}$, so the correct option is C.

Would you like further explanation or details? Here are related questions:

1. How is the Pythagorean theorem applied to other geometric shapes?
2. Can we use the same method for a square? Why?
3. What is the formula for the perimeter of a rectangle?
4. How can we find the area of rectangle $PQRS$?
5. How does changing the dimensions affect the diagonal length?

Tip: Always check if a rectangle's dimensions fit the Pythagorean theorem before calculating its diagonal.