For real numbers a and b, define a b = √a^2 + b^2. What is the value of (5 12) ((-12) (-5))?

2005 AMC 10B Problem 4

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For real numbers

a

and

b,

define

a \diamond b = \sqrt{a^2 + b^2}.

What is the value of

(5 \diamond 12) \diamond ((-12) \diamond (-5))?

0

\dfrac{17}{2}

13

13\sqrt{2}

26

Evaluate the two inner operations first

Both

5\diamond12

and

(-12)\diamond(-5)

equal

13

Each inner expression evaluates to

5 \diamond 12 = \sqrt{5^2 + 12^2} = 13

and similarly

(-12) \diamond (-5)

= \sqrt{144 + 25} = 13.

Then

13 \diamond 13 = \sqrt{13^2 + 13^2} = 13\sqrt{2}.

Thus, D is the correct answer.