24.28 units
Explanation
Perimeter is the distance around the edge of a shape, to find the perimeter of the given figure we can use the distance bewteen 2 points formula,it says
[tex]\begin{gathered} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ where \\ P1(x_1,y_1) \\ P2(x_2,y_2) \end{gathered}[/tex]
so
Step 1
identify the vertices ;
[tex]\begin{gathered} A(-2,3) \\ B(0,6) \\ C(5,4) \\ D(4,-1) \\ E(-1,-2) \end{gathered}[/tex]
Step 2
now, find the length of each segment
a)AB
replace in the formula
[tex]\begin{gathered} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ d_{AB}=\sqrt{(0-(-2))^2+(6-3)^2} \\ d_{AB}=\sqrt{4+9} \\ d_{AB}=\sqrt{13} \end{gathered}[/tex]
b)BC
[tex]\begin{gathered} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ d_{BC}=\sqrt{(5-0)^2+(4-6)^2} \\ d_{BC}=\sqrt{25+4} \\ d_{BC}=\sqrt{29} \end{gathered}[/tex]
c)CD
[tex]\begin{gathered} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ d_{CD}=\sqrt{(4-5)^2+(-1-4)^2} \\ d_{CD}=\sqrt{1+25} \\ d_{CD}=\sqrt{26} \end{gathered}[/tex]
d)DE
[tex]\begin{gathered} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ d_{DE}=\sqrt{(-1-4)^2+(-2-(-1))^2} \\ d_{DE}=\sqrt{25+1} \\ d_{DE}=\sqrt{26} \end{gathered}[/tex]
e) EA
[tex]\begin{gathered} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ d_{EA}=\sqrt{(-2-(-1))^2+(3-(-2))^2} \\ d_{EA}=\sqrt{1+25} \\ d_{EA}=\sqrt{26} \end{gathered}[/tex]
Step 3
finally, the perimeter is the sum of the sides length, so
[tex]\begin{gathered} perimeter=AB+BC+CD+DE+EA \\ replacing \\ perimeter=\sqrt{13}+\sqrt{29}+\sqrt{26}+\sqrt{26}+\sqrt{26} \\ perimeter=24.28 \end{gathered}[/tex]
so, the perimeter is 24.28 units
