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Bottom two paragraphs:
Let's say the sides can be represented in terms of x, 2x, and x*sqrt3.
Now, if they tell us that the side opposite the 60 is 27, and we know that this 27 represents x*sqrt3, then we can find x from there --
x*sqrt3 = 27, so x = 27/sqrt3
The GMAT doesn't like radicals in denominators, so we multiply by sqrt3/sqrt3 (It's equal to 1, so it doesn't change the value) and we get 27sqrt3 / 3, which is 9sqrt3.
(I think this answers statement 1, though I believe you may have meant 27sqrt3 rather than 29sqrt3)
The second triangle (bottom paragraph) works the same way.
The side opposite the 60 degree angle is sqrt48, and is also x*sqrt3 in our "pattern" triangle.
x*sqrt3 = sqrt48
Divide both sides by sqrt3 and we have
x = sqrt48 / sqrt3
Remember that sqrt(a) / sqrt(b) = sqrt(a/b), so
x = sqrt (48/3) which is sqrt(16) or 4.
This tells us that the value of x is 4, so the hypotenuse, which is 2x in our "pattern" is 8.
As for your final question -
If we know a triangle has one side equal to 5 and it's the shortest side, there's no way we can determine the other two sides without knowing more about the triangle. If we know it's a right triangle, it MIGHT be a 5-12-13 triangle, but there are many many triangles, even right triangles, that can be formed with 5 as the shortest side.
Take 5 and 6 as the perpendicular sides, for example -- the third side can be found using the Pythagorean Theorem, and is sqrt(61).
If, on the other hand, we know that the angles in the triangle are all the same as the angles in a 5-12-13 triangle (in the case of similar triangles), and we know that the shortest side is 5, then we DO know the other two sides are 12 and 13. We must know a fair amount about the triangle to determine this, though.
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