### Instruction: Octagon $ABCDEFGH$ is equiangular. Given that $AB=1$, $BC=2$, $CD=3$, $DE=4$, and $EF=FG=2$, compute the perimeter of the octagon.### Response: I'm going to start by drawing a picture of the octagon and labeling the sides as given.
<diagram>
Since the octagon is equiangular, I know that all the interior angles are equal to 135 degrees.
This means that I can divide the octagon into eight isosceles triangles by drawing the diagonals from the center to each vertex.
<diagram>
Now, I can use the base angles of these triangles to find the lengths of the remaining sides.
For example, in triangle AOH, I have that angle OAH = angle OHA = (180 - 135)/2 = 22.5 degrees.
So, using the sine rule, I have that

sin(22.5)/OH = sin(135)/1
Solving for OH, I get that OH = sin(22.5)/sin(135) = 0.414.
Similarly, I can find that OG = sin(22.5)/sin(135) = 0.414.
Now, I can use the same logic to find the lengths of the other sides.
In triangle BOC, I have that angle OBC = angle OCB = (180 - 135)/2 = 22.5 degrees.
So, using the sine rule, I have that

sin(22.5)/OB = sin(135)/2
Solving for OB, I get that OB = 2sin(22.5)/sin(135) = 0.828.
Similarly, I can find that OD = 2sin(22.5)/sin(135) = 0.828.
In triangle COD, I have that angle OCD = angle ODC = (180 - 135)/2 = 22.5 degrees.
So, using the sine rule, I have that

sin(22.5)/OC = sin(135)/3
Solving for OC, I get that OC = 3sin(22.5)/sin(135) = 1.242.
Similarly, I can find that OE = 3sin(22.5)/sin(135) = 1.242.
In triangle DOE, I have that angle ODE = angle OED = (180 - 135)/2 = 22.5 degrees.
So, using the sine rule, I have that

sin(22.5)/OD = sin(135)/4
Solving for OD, I get that OD = 4sin(22.5)/sin(135) = 1.656.
Similarly, I can find that OF = 4sin(22.5)/sin(135) = 1.656.
Now, I have found all the lengths of the sides of the octagon, so I can simply add them up to get the perimeter.
The perimeter is

AB + BC + CD + DE + EF + FG + GH + HA

= 1 + 2 + 3 + 4 + 2 + 2 + (OH + OG) + (OB + OD)

= 14 + 0.828 + 0.828 + 1.242 + 1.242 + 1.656 + 1.656

= 21.452
Therefore, the perimeter of the octagon is 21.452.