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

2005 AMC 10B Problem 4 — Solution

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For real numbers $a$ and $b,$ define $a \diamond b = \sqrt{a^2 + b^2}.$ What is the value of $(5 \diamond 12) \diamond ((-12) \diamond (-5))?$

$0$

$\dfrac{17}{2}$

$13$

$13\sqrt{2}$

$26$

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.