The length the sides of an isosceles right triangle look like this:


So the given information is telling us that: \(x + x + x\sqrt{2} = 16 + 16 \sqrt{2}\)

At this point, it's probably fastest to start testing some answer choices.
Notice that if \(x=8\), we get: \(8 + 8 + 8\sqrt{2} = 16 + 16 \sqrt{2}\)
Simplify to get: \(16 + 8\sqrt{2} = 16 + 16 \sqrt{2}\)

This result tells us two things:
1) \(x=8\) is NOT a solution to the above equation, since \(16 + 8\sqrt{2} \neq 16 + 16 \sqrt{2}\)

2) Since \(16 + 8\sqrt{2} < 16 + 16 \sqrt{2}\), the actual solution to the equation must be GREATER THAN 8.
This means we can eliminate A, B and D

Let's now test \(x = 8\sqrt{2}\)
We get: \(8\sqrt{2} + 8\sqrt{2} + (8\sqrt{2})(\sqrt{2}) = 16 + 16 \sqrt{2}\)
Simplify: \(16\sqrt{2} + 16 = 16 + 16 \sqrt{2}\)

Perfect!

Answer: C