Answer:
The perimeter of △LMN is 8 + [tex]\sqrt{10}[/tex]
Step-by-step explanation:
Step 1: Finding the length of LM
Distance formula = [tex]\sqrt{(x_2-x_1)^2 +(y_2-y_1)^2}[/tex]
here
[tex]x_1[/tex]= 2
[tex]x_2[/tex]= -2
[tex]y_1[/tex]= 4
[tex]y_2[/tex]=1
LM = [tex]\sqrt{(-2-2)^2 +(1-4)^2}[/tex]
LM = [tex]\sqrt{(-4)^2 +(-3)^2}[/tex]
LM = [tex]\sqrt{16 +9)}[/tex]
LM = [tex]\sqrt{25}[/tex]
LM = 5
Step 2: Finding the length of MN
Distance formula = [tex]\sqrt{(x_2-x_1)^2 +(y_2-y_1)^2}[/tex]
here
[tex]x_1[/tex]= -2
[tex]x_2[/tex]= -1
[tex]y_1[/tex]= 1
[tex]y_2[/tex]= 4
LM = [tex]\sqrt{(-1-(-2))^2 +(4 - 1)^2}[/tex]
LM = [tex]\sqrt{(11+2)^2 +(3)^2}[/tex]
LM = [tex]\sqrt{1 +9)}[/tex]
LM = [tex]\sqrt{10}[/tex]
Step 3 : Finding the length of NL
Distance formula = [tex]\sqrt{(x_2-x_1)^2 +(y_2-y_1)^2}[/tex]
here
[tex]x_1[/tex]= -1
[tex]x_2[/tex]= 2
[tex]y_1[/tex]= 4
[tex]y_2[/tex]= 4
NL = [tex]\sqrt{(2-(-1))^2 +(4 - 4)^2}[/tex]
NL = [tex]\sqrt{(3)^2 +0}[/tex]
NL = [tex]\sqrt{9 +0}[/tex]
B = [tex]\sqrt{9 }[/tex]
NL = 3
Step 4: Finding the perimeter of the triangle
Perimeter = length of LM + length of MN + length of NL
Perimeter = 5 + [tex]\sqrt{10}[/tex] + 3
Perimeter = 8 + [tex]\sqrt{10}[/tex]
