Quandaries
and Queries 

Please could you tell me why for my coursework (where I have to find the largest area that a fence 1000m long can cover) why I should only test equilateral and isoceles triangles? We were told NOT to do right angled triangles but I was wondering why not? Thankyou
very much 

Hi Petey Suppose that triangle ABC has a right angle at C and that the perimeter is 1000 m. Let M be the midpoint of BC. Construct an isosceles triangle PBC on the same base, BC, and with CA = PM. The triangles ABC and PBC have the same base and height and hence the same area. I want to show you that the perimeter of triangle PBC is less than the perimeter of triangle ABC.
Claim: PB + CP = 2 PB < AB + CA.
Thus the perimeter of triangle PBC is less than the perimeter of triangle ABC. Hence if you move P upwards to the place where the triangles have the same perimeter you have a new isoscelese triangle PBC with the same primeter as ABC but a larger area. Cheers, 

