Answer:
Perimeter of rectangle= [tex]2 (\sqrt{10} + \sqrt{29})[/tex]
= [tex]2\sqrt{10} + 2\sqrt{29}[/tex].
Step-by-step explanation:
Given: [tex]E(x_{1}, y_{1}) = (1. -1), F(x_{2}, y_{2}) = (-4, 1), G(x_{3}, y_{3}) = (-3, 4), H(x_{4}, y_{4}) = (2, 2)[/tex]
Using distance formula:
Length of EF = [tex]\sqrt{(x_{2} - x_{1} )^2 + (y_{2} - y_{1})^2}[/tex]
= [tex]\sqrt{(-4 - 1)^2 + (1 - (-1))^2}[/tex]
= [tex]\sqrt{(-4 - 1)^2 + (1 + 1))^2}[/tex]
= [tex]\sqrt{(-5)^2 + (2))^2}[/tex]
= [tex]\sqrt{25 + 4}[/tex]
= [tex]\sqrt{29}[/tex]
Length of FG = [tex]\sqrt{(x_{3} - x_{2} )^2 + (y_{3} - y_{2})^2}[/tex]
= [tex]\sqrt{(-3 - (-4)^2 + (4 - 1)^2}[/tex]
= [tex]\sqrt{(1)^2 + (3)^2}[/tex]
= [tex]\sqrt{1 + 9}[/tex]
= [tex]\sqrt{10}[/tex]
Length of GH = [tex]\sqrt{(x_{4} - x_{3} )^2 + (y_{4} - y_{3})^2}[/tex]
= [tex]\sqrt{(2 - (-3))^2 + (2 - 4)^2}[/tex]
= [tex]\sqrt{(5)^2 + (-2))^2}[/tex]
= [tex]\sqrt{25 + 4}[/tex]
= [tex]\sqrt{29}[/tex]
Length of HE = [tex]\sqrt{(x_{4} - x_{1} )^2 + (y_{4} - y_{1})^2}[/tex]
= [tex]\sqrt{(2 - 1)^2 + (2 - (-1))^2}[/tex]
= [tex]\sqrt{(1)^2 + (3)^2}[/tex]
= [tex]\sqrt{1 + 9}[/tex]
= [tex]\sqrt{10}[/tex]
∵ EFGH is a rectangle ∴ EH = FG and EF = HG
Perimeter of rectangle = 2 ( EF + FG + GH + HE)
= 2 (EF + FG)
= [tex]2 (\sqrt{10} + \sqrt{29})[/tex]
= [tex]2\sqrt{10} + 2\sqrt{29}[/tex]
Therefore option (b) is the correct answer.
