It's interesting that areas from coordinates are much simpler than perimeters from coordinates. It's almost like math is telling us area is more fundamental than length.
p = AB+BC+CD+DE+EA
We use the Pythagorean theorem, square root form, on all the sides. We can skip a lot of the square roots because most of these are the hypotenuse of a 3/4/5 right triangle.
B-A=(4-0, 4-1)=(4,3) so length AB=5
C-B=(7-4,0-4)=(3,-4), so again BC=5
D-C=(3-7, -3-0)=(-4,-3) so again CD=5
E-D=(-4-3, -4 - -3) = (-7, -1). Ends our streak. DE=√(7²+1²)=√50=5√2.
A-E=(0 - -4,1 - -4)=(4,5) so EA=√(4²+5²)=√41.
Three of five were 3/4/5.
p = 5+5+5+5√2+√41 = 15+5√2+√41
I hate to ruin a nice exact answer with an approximation.
p ≈ 28.474192049298324
Answer: 28.47, last choice
