If there's a quadrilateral ABCD, and the sides AB + BC + CD add up to 24, how can I find the maximum area of the quadrilateral formed, what length should the four sides of the quadrilateral be, and what should the interior angles be?
I think Lagrange multiplier needs to be used for such a case, but I'm not sure how to start. I think the constraint will be x + y + z = 24, but I'm not sure how to represent the area function. Any help is greatly appreciated!
On
Reflect the quadrilateral across the fourth side to produce a hexagon with perimeter $48$. Of all hexagons with a given perimeter, the one with the greatest area is the regular hexagon (proof). So the maximum area of the hexagon is $96\sqrt3$, and so the maximum area of the quadrilateral is $48\sqrt3$.
Denote the four sides of the quadrilateral by $w = DA$, $x = AB$, $y = BC$, $z = CD$.
Let $d = AC$ be the length of one diagonal. This splits the quadrilateral into two triangles, $\triangle ABC$ and $\triangle ACD$.
Use Heron's formula to find the area of the two triangles.
$$A = \frac{1}{4}\sqrt{(x + y + d)(-x + y + d)(x - y + d)(x + y - d)} + \frac{1}{4}\sqrt{(w + z + d)(-w + z + d)(w - z + d)(w + z - d)}$$
But we're given that $x + y + z = 24$, so:
$$A = \frac{1}{4}\sqrt{(x + y + d)(-x + y + d)(x - y + d)(x + y - d)} + \frac{1}{4}\sqrt{(w + z + (24 - x - y))(-w + (24 - x - y) + d)(w - (24 - x - y) + d)(w + (24 - x - y) - d)}$$
This gives us a formula for the area in terms of three sides of the quadrilateral ($w$, $x$, and $y$) and the diagonal $d$. Some work will needed to be done to find the maximum, but at least now we know what to maximize.