To find the perimeter of the polygon defined by the points A, B, C, D, E, and F, we first calculate the lengths of each side using the distance formula. The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here are the coordinates for each point:
- A: (2, -6)
- B: (-5, -6)
- C: (-5, 6)
- D: (6, 6)
- E: (6, 1)
- F: (2, 1)
Let's calculate the distances between each pair of consecutive points, starting from A and going around the polygon:
-
AB: \[ d_{AB} = \sqrt{((-5) - 2)^2 + ((-6) - (-6))^2} = \sqrt{(-7)^2 + 0^2} = \sqrt{49} = 7 \]
-
BC: \[ d_{BC} = \sqrt{((-5) - (-5))^2 + (6 - (-6))^2} = \sqrt{0^2 + (12)^2} = \sqrt{144} = 12 \]
-
CD: \[ d_{CD} = \sqrt{(6 - (-5))^2 + (6 - 6)^2} = \sqrt{(11)^2 + 0^2} = \sqrt{121} = 11 \]
-
DE: \[ d_{DE} = \sqrt{(6 - 6)^2 + (1 - 6)^2} = \sqrt{0^2 + (-5)^2} = \sqrt{25} = 5 \]
-
EF: \[ d_{EF} = \sqrt{(2 - 6)^2 + (1 - 1)^2} = \sqrt{(-4)^2 + 0^2} = \sqrt{16} = 4 \]
-
FA: \[ d_{FA} = \sqrt{(2 - 2)^2 + (-6 - 1)^2} = \sqrt{0^2 + (-7)^2} = \sqrt{49} = 7 \]
Now we sum all the side lengths to find the perimeter \(P\):
\[ P = d_{AB} + d_{BC} + d_{CD} + d_{DE} + d_{EF} + d_{FA} \] \[ P = 7 + 12 + 11 + 5 + 4 + 7 = 46 \]
Thus, the perimeter of the polygon is \(46\) kilometers.
So the answer is 46 kilometers.