The octagon ABCDEFGH is inscribed in a circle, with the vertices around the circumference in the given order. Given that the polygon ACEG is a square of area 5 and the polygon BDFH is a rectangle of area 4, find the maximum possible area of the octagon.
Since ACEG is a square with area 5, diameter AE=sqrt(10).
Since BDFH is a rectangle and BF=sqrt(10) set DF=4/BD, then use Pyth to get an easily soluble double quadratic and find BD=2sqrt(2) and DF=sqrt(2).
On the assumption of symmetry (BD parallel to AE) we're set.
The area of the octagon is twice the area of the pentagon ABCDE, which is comprised of trapezoid ABDE and triangle BCD.
ABDE has parallel sides AE=sqrt(10) and BD=2sqrt(2) and height DF/2 = sqrt(2)/2.
BCD has base BD and altitude = sqrt(10)/2  sqrt(2)/2.
Do the arithmetic and find area octagon = 3*sqrt(5)

Posted by xdog
on 20121128 22:01:43 