ABCD is a square of side length 1. E , F , G and H are the midpoints of AB , BC , CD and DA , respectively. The lines FA , AG , GB , BH , HC , CE , ED and DF determine a convex 8-gon. By symmetry, this octagon has equal sides. If s is the side length of the octagon, then s 2 can be expressed as a b , where a and b are coprime positive integers. What is the value of a+b ?

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To find the side length of the octagon formed by the lines FA, AG, GB, BH, HC, CE, ED, and DF, we can break down the process step by step.

1. Start by drawing the square ABCD and marking the midpoints of each side as E, F, G, and H.
2. Connect the midpoints of each side to their adjacent vertices to form the octagon.
3. Notice that the octagon is formed by alternating segments of length s/2 and s, where s is the side length of the square.
4. Since the square has a side length of 1, the segments EF and GH have a length of 1/2, and the segments FA/AG/GB/BH/HC/CE/ED/DF have a length of 1.
5. Now, let's look at the perimeter of the octagon. It consists of 8 sides, each with length s/2 or s.
6. The perimeter of the octagon is equal to the sum of the lengths of all 8 sides. This can be expressed as: (4 * s/2) + (4 * s), which simplifies to 3s + 2s = 5s.
7. Now, we know that the perimeter of the octagon is 5s.
8. To find the side length s, we need to find the perimeter of the octagon. But since it is a square, the perimeter is also equal to 4 times its side length. Therefore, 5s = 4s.
9. Solving for s, we get s = 0.
10. Now, we want to find s^2. Substituting the value of s, we have (0^2) = 0.
11. Therefore, a = 0 and b = 1.
12. Finally, a + b = 0 + 1 = 1.

So, the value of a + b is 1.