To find the length of the diagonal cut, which is the hypotenuse of a right triangle, we can use the Pythagorean theorem. Given that the length ($a$) and width ($b$) of the rectangle are 5 inches and 12 inches respectively, we can calculate the diagonal ($c$) as follows:

$c = \sqrt{a^2 + b^2}$

Substituting the given values:

$c = \sqrt{5^2 + 12^2}$ $c = \sqrt{25 + 144}$ $c = \sqrt{169}$ $c = 13$

Therefore, the length of the diagonal cut that Nick made is 13 inches.

Would you like more details on this solution, or do you have any questions?

Here are some related questions to expand understanding:

1. What would the diagonal be if the dimensions of the paper were doubled?
2. How does the Pythagorean theorem apply to other shapes, like cubes?
3. What if Nick cut the paper from a different corner? Would the length change?
4. How would you find the area of each triangle formed?
5. What if the paper had a different shape, like a square—how would you approach it?

Tip: The Pythagorean theorem is a powerful tool for solving any right triangle problem, and it applies in many practical scenarios!