Find the dimensions and orientation of the largest area (a) square and (b) rectangle you can draw inside an equilateral triangle of side 1
I took rectangle to mean nonsquare.
Square
A square side=x aligned with one side on the base of the triangle and the other corners touching the other sides gives a 306090 triangle with the vertical side=sqrt(3)*(1x)/2. But this side=x and x = sqrt(3)/(2 + sqrt(3)) = 2*sqrt(3)  3 and area = .2153903091+
Rectangle.
x=width, y=heighth. Using the same 306090 triangle, y = sqrt(3)*(1x)/2 and area xy = x*sqrt(3)*(1x)/2 which is minimum when x = 1/2, so y=sqrt(3)/4, area = sqrt(3)/8 = .2165063509+

Posted by xdog
on 20131118 19:33:32 