INSEADIESE wrote:
now 8 units is the longest side of the triangle, side opposite to 108 degree angle, each of the other two equal sides , sides opposite to 36 degree angles, shall be less than 8.. this is it.. all I can infer is that each side of the pentagon is less than 8 units, which will give me a perimeter both greater and smaller than 26 units hence this option is not sufficient.
There are a couple of issues in your post. If you draw a line of length 8, and then from either end draw two lines at exactly 36 degrees to make a triangle, there's only one triangle you could possibly draw. Once you decide on one length and all of your angles in a triangle, the lengths of the other two sides are completely determined - they can't vary. With these specific angles (36-36-108), it's not easy to find the precise lengths of the sides (which is why this question is so hard), but it is possible to solve for them.
For the GMAT, test takers typically learn that when you have 30-60-90 triangles, or 45-45-90 triangles, then when you know one side, you can find all three sides. There's actually nothing special about those angles. One thing you learn in trigonometry is that if you know all three angles in a triangle, and you know one side, it is always possible to find all three sides, no matter what the angles are. So if you know trigonometry, if you had a triangle with angles of 87, 52 and 41 degrees, say, and you knew the longest side had length 10, you'd be able to find the other two side lengths.
Fortunately you never need to do that on the GMAT, except when you have very simple angles like 30-60-90, because you never need to use sines and cosines on the GMAT. But your assumption, that the equal sides in the 36-36-108 triangle in this question can have various lengths, is not correct, once you choose a length for the long side. And that's why the answer does not need to be E for this question (in fact it's not E -- the answer is D).
INSEADIESE wrote:
Statement 2– this basically is the same as statement 1 ie length of each diagonal is 8 unites ie the radius is 4 .... a
I think you're assuming the diagonals of the pentagon are also diameters of the circle, and that's not the case. The diagonals of the pentagon are shorter than the diameter of the circle.
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