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Great question, Surajalok - this brings up a crafty GMAT writing style that I've always admired (while watching students struggle with it).
The perimeter of an isosceles right triangle can be calculated using the fact that the ratio of the sides will be x, x, and x*sqrt 2. So, the perimeter will be 2x + x*sqrt 2. We can set that equal to the given perimeter:
2x + x*sqrt 2 = 16sqrt 2 + 16
This one is probably more difficult to solve algebraically than conceptually, so let's just look at the concept. We know that one of the terms (either 16 or 16sqrt 2) will be divided into the two shorter sides, and the other term will represent the longer side (the hypotenuse). So, our options are:
16 represents the two smaller sides; 16sqrt2 represents the long side
OR
16sqrt 2 represents the two smaller sides; 16 represents the long side
If we try the first option, we'd cut 16 in half to create two sides, and our sides would be 8, 8, and 16sqrt 2. This doesn't follow the ratio - the third side, if the first two are both 8, should be 8sqrt 2, so this one doesn't hold.
Trying the second one, we find that the shorter sides are 8sqrt 2 and the long side would be 16. Well, 8sqrt 2 * sqrt 2 fits the ratio, and gives us 8*2 = 16, so the long side fits with the shorter ones.
Therefore, the hypotenuse in this problem has a length of 16.
What I love about this problem (and the many like it) is that the authors of the test know that, because of the ratio x-x-x*sqrt 2, we like to see our hypotenuses end with a square root of 2 on the end. However, if the shorter sides carry a sqrt 2, then the ratio will multiply it out of the hypotenuse, which will then be an integer. It doesn't seem to look right when you're viewing the answer choices, but the math holds up. So...be careful when using right triangle side ratios (45-45-90 and 30-60-90) that you don't preemptively commit to having the radical sign assigned to the side that carries it in the ratio - the GMAT is great at creatively moving the radical to throw you off!
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