erjamit wrote:If a, b and c are sides of a triangle with c as the hypotenuse then,
a^2+b^2 = c^2, and we have the right triangle with say angle x = 90 (x is angle opposite to hypotenuse.
Now can we deduce the following also
if a^2+b^2< c^2, x > 90, and if a^2+b^2 > c^2, x < 90.
Yes, you can. You can see this visually: imagine a right angle triangle, with legs a and b, and hypotenuse c. We know that a^2 + b^2 = c^2. Imagine rotating just side a, so that the 90 degree angle gets larger, without changing the length of a and b. You'd need to make c longer to complete the triangle, and we haven't changed a^2 + b^2, so now a^2 + b^2 < c^2. If you rotate a in the other direction, so the angle gets smaller than 90 degrees, you'd need to make c smaller to draw the triangle.
You can also see why this is true from the Cosine Law, from trigonometry (which you don't need to know for the GMAT!). In any triangle with sides a, b and c, if C is the angle opposite the side of length c, then
c^2 = a^2 + b^2 - 2ab*cos(C)
When C = 90, cos(90) = 0, and we get the Pythagorean Theorem.
When C < 90, cos(C) is positive, so c^2 < a^2 + b^2.
When C > 90, cos(C) is negative, so c^2 > a^2 + b^2.
I glanced at that thread, and there's some dodgy math there, so I wouldn't recommend reading it too closely. For example, someone says that in right angled triangles, the sides are always in a 3:4:5 ratio, which is absolutely untrue.
