find area of Triangle ABF

Question

In the figure given below, rectangle $CDEF$ with perimeter $32$ has the maximum area. Find the area of the triangle $ABF$

So, I tried the following

$P = 2W+2H$ where $P$ is given $32$. I am not able figure out which will be next step & how can I find Area of Triangle $ABF$

Answer 1

Here the rectangle with perimieter $32$ has maximum area.Using calculus (or any other method) that it must be a rectangle.

As perimeter is $32$ the each side length is $8$ . so now each side is $8$ units. So Cf= $8$ units . As triangle BFC IS Isosceles with right angle, so Side BF = $8$ now triangle ABF is 30-60-90 triangle so side BA = BF $\times \sqrt {3}$

Now as we know area of this triangle = $\frac {1}{2} \times BA \times BF$ = $\frac {1}{2} \times 8 \times \sqrt {3} \times 8$ = $32 \sqrt {3}$ units