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[tex]\huge \boxed{\sf 8+4\sqrt{5} }\\\\\\\sf Using\ the\ distance\ formula \\\\\displaystyle d = \sqrt{(x_2 - x_1)^2 + (y_2-y_1)^2}[/tex]

[tex]\sf A(0,4)\ and\ B(2,0)\\\\d = \sqrt{(2 - 0)^2 + (0-4)^2}=2\sqrt{5} \\\\B(2,0)\ and\ C(2,-2)\\\\d = \sqrt{(2 - 2)^2 + (-2-0)^2}=2 \\\\C(2,-2)\ and\ D(0,-2)\\\\d = \sqrt{(0 - 2)^2 + (-2-(-2))^2}=2 \\\\D(0,-2)\ and\ E(-2,2)\\\\d = \sqrt{(-2 - 0)^2 + (2-(-2))^2}=2\sqrt{5} \\\\E(-2,2)\ and\ F(-2,4)\\\\d = \sqrt{(-2 - (-2))^2 + (4-2)^2}=2\\\\F(-2,4)\ and\ A(0,4)\\\\ d=\sqrt{(0-(-2))^2+(4-4)^2} =2[/tex]

[tex]\sf Adding\ all\ the\ distances \\\\2\sqrt{5} +2+2+2\sqrt{5}+2+2 =8+4\sqrt{5} =16.94427190...[/tex]

mhanifa

Answer:

  • The perimeter is 8 + 4√5 units

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Four of the segments are horizontal or vertical with the length of 2 units:

  • FA = FE = BC = DC = 2

Two of the segments have same length (parallel segments) with 4 units vertical and 2 units horizontal difference of coordinates:

  • AB = ED = √(4² + 2)² =√20 = 2√5

The perimeter is:

  • P = 4*2 + 2*2√5 = 8 + 4√5

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