There's a pentagon, not necessarily regular, with integral sides, and a point within it from which lines are drawn to each of the pentagon's vertices. Each of these five lines from the internal point is a different integral length less than 25, and their angles with each other at that internal point are an integral number of degrees, none of which is 90.

How long are the lines and the sides of the pentagon? What is its total perimeter?

Where a, b, and c are the sides of a triangle, and a and b are both integers less than 25, the angle opposite a must be either 60, 90, or 120 degrees.
As it is given that none of the angles is 90, and the five internal angles must sum to 360, four of the angles must be 60 degrees and one must be 120 degrees.
There are ten triangles where the two sides adjacent to the 60 degree angle are integer lengths less than 25 with the side opposite the 60 degree angle is also an integer length:

 3   8   7
 5   8   7
 5  21  19
 6  16  14
 7  15  13
 8  15  13
 9  24  21
10  16  14
15  24  21
16  21  19

As each side must occur in another of these triangles, there are only seven different pentagons that have four of these triangles. The five lengths of each of these are:

 3 -  8 -  5 - 21 - 16
 5 -  8 - 15 - 24 -  9
 6 - 16 - 21 -  5 -  8
 7 - 15 -  8 -  5 - 21
10 - 16 - 21 -  5 -  8
16 - 21 -  5 -  8 - 15
21 -  5 -  8 - 15 - 24

Only one, 21-5-8-15-24, with an angle of 120 between the two "end" sides, 21 and 25, provide the final side of integer length.

The perimeter of the pentagon -- the sum of the length of side c of each of the five triangles -- is, then, 19 + 7 + 13 + 21 + 39 = 99.

Posted by Dej Mar on 2009-06-09 23:03:42