What is the maximum area that can be enclosed by 4 straight pieces of fencing of length 2, 6, 7 and 9 yards?

Here are a couple of figures ... I don't know the most general or complete solution.

A (9,6,7) triangle has area = 20.98, which gives a baseline to start from.

Then use the 2 piece to open it up some.

I tried triangles by attaching the 2 to the other pieces to make one long piece,

and using Heron's formula, got the following results:

(9+2), 7, 6 = 18.97 -- not even as good as just the (9,7,6) triangle.

(7+2), 6, 9 = 25.46

(6+2), 7, 9 = 26.83

As usual, the figure with the least differences among the sides had the largest area.

Then I tried a trapezoid, putting the the 2 opposite the 9, and parallel.

So we have a figure like this:

........._2_

......./|......|.\

......6.| .....|..7

...../_|____|__\

......x ...2....z <--- 9 long side here

Let the height be y

Then x + z = 7

x^2 + y^2 = 36

z^2 + y^2 = 49

z^2 - x^2 = 13

(7-x)^2 - x^2 = 13

49 - 14x + x^2 - x^2 = 13

14x = 36

x = 36/14 = about 2.57

z = 7 - 36/14 = 62/14 = about 4.43

7056/14^2 - 36^2/14^2 = y^2

y^2 = 5760/14^2 = about 29.39

y = 5.42

Area = y (2 + 9)/2 = 5.42 * 5.5 = 29.82

That's my best try - I imagine it's either the maximum,

or fairly close to it.