Let me try.
First, draw lines from the center of pentagon ( and a circle) to its vertices. The lines you draw are equal to radius of the circle a pentagon inscribed in. We have 5 equal triangles now.
the angle between every radius is 360/5=72. Sum of other two angles of every triangle are (180-72)=108. Since every triangle has 2 equal sides, such triangle is isosceles. Hence, the angles of a triangle are 72, 54, 54. Note that the angle of 72 degrees is opposite to the side a pentagon.
We only need to know if the perimeter of a pentagon is >26. So, knowing the relationship among 3 sides of a triangle ( 2 of which are radiuse) we could know the range of a third side of a triangle ) or a side of a pentagon)
(1) given that radius=4
So the side of a triangle (x) must be: x<8 (4+4), 4+x>4=>x>0, and also must be larger than 2 other sides of a triangle (since it has the opposite angle larger)==> x>4, so X:(4;8), the least possible perimeter might be 4.1*5, 5*5=25, 5*6=30. 20,5<26, 25<26, 30>26. Insufficient.
(2) diagonal <8. (really need to have a picture)
When you draw whatever diagonal of our regular pentagon you get a isosceles triangle as well, following the same rules as in the (1) we could get a range of a sides - since the angle of 108 degrees is opposite to the diagonal ==> diagonal of a pentagon is the largest side of a triangle. diagonal is <8. then 2x>8, x>4. and x+8>x, x>0.
If I have not forgotten something. It seems to me like the answer is E. But i clearly don't remember some properties of the regular pentagon. hehe
Good DS question. If someone could remind what am i missing i ll be really pleased )))
Last edited by
NikolayZ on Wed Oct 07, 2009 11:54 am, edited 2 times in total.
