I love this problem...I tell my students all the time to "think like the testmaker" and to try to predict how the GMAT can make a concept like "sides of an isosceles right triangle" harder. This one nails it...and I think you can get inside the mind of whoever wrote it.
Hopefully we all know by now that in a 45-45-90 triangle the side ratios will be x - x - x(sqrt2). So WHY are they asking this question? They're NOT testing whether you know that formula...there's something bigger that they're testing!!
How can they make this harder? Your mind wants to see the sqrt2 on the hypotenuse side. That's just where it makes sense upon a quick glance. But what if the long side is the integer and the other sides have the square root on them? That makes it a much harder problem - the correct answer looks unexpected. And in doing so the GMAT can test whether you're likely to jump to conclusions based on gut feel, or whether you'll really break down and follow the data even if it seems counter-intuitive. They're testing whether you can double-check and make sense of information that looks awkward but is, indeed, valid.
Because any side could include a radical - the radical sign around "sqrt 2" is a RATIO, but not a requirement for the hypotenuse, we need to go back to the ratio. The perimeter of this triangle will equal:
x + x + x*sqrt 2
so, 16 + 16*sqrt 2 = 2x + xsqrt 2
At this point, the algebra may end up being messier than just plugging in. Hopefully you can see that there's an imbalance, though - if x is 16, then 2x is 32, so you'd have:
16 + 16sqrt2 = 32 + 16sqrt 2
x can't be any integer, or this situation will occur. So the other choices, those that include a square root of 2 for x, have to hold:
x = 8sqrt 2 gives us:
16 + 16sqrt 2 = 2(8sqrt 2) + 8sqrt 2 * sqrt 2
16 + 16sqrt 2 = 16 sqrt 2 (we have one match!) + 8*2 (which is 16...there's our other match!)
So if x = 8sqrt 2, then the hypotenuse - x multiplied by sqrt 2 - becomes 8sqrt2*sqrt2 = 8*2 = 16.
The algebraic/mathematic proof is a little long-winded, but the takeaway is big. BEWARE the right-triangle side ratio (45-45-90 or 30-60-90) that assigns the square root radical to a side DIFFERENT FROM the one you'd expect to carry it. The math STILL works, but because it's so counter-intuitive it will trap and/or confuse a whole bunch of test takers. That's what they're testing on this.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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