Answer:
11.4 units
Step-by-step explanation:
To find the perimeter of triangle CDE, we need to calculate the lengths of the three sides CD, DE, and EC using the distance formula:
[tex]\boxed{\begin{array}{l}\underline{\sf Distance \;Formula}\\\\d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\\\\textsf{where:}\\ \phantom{ww}\bullet\;\;d\;\textsf{is the distance between two points.} \\\phantom{ww}\bullet\;\;\textsf{$(x_1,y_1)$ and $(x_2,y_2)$ are the two points.}\end{array}}[/tex]
Given vertices:
Therefore:
[tex]CD=\sqrt{(x_D-x_C)^2+(y_D-y_C)^2}\\\\CD=\sqrt{(2-1)^2+(-1-2)^2}\\\\CD=\sqrt{1^2+(-3)^2}\\\\CD=\sqrt{1+9}\\\\CD=\sqrt{10}[/tex]
[tex]DE=\sqrt{(x_E-x_D)^2+(y_E-y_D)^2}\\\\DE=\sqrt{(-2-2)^2+(-1-(-1))^2}\\\\DE=\sqrt{(-4)^2+0^2}\\\\DE=\sqrt{16}\\\\DE=4[/tex]
[tex]EC=\sqrt{(x_C-x_E)^2+(y_C-y_E)^2}\\\\EC=\sqrt{(1-(-2))^2+(2-(-1))^2}\\\\EC=\sqrt{3^2+3^2}\\\\EC=\sqrt{9+9}\\\\EC=\sqrt{18}\\\\EC=3\sqrt{2}[/tex]
Now, we can find the perimeter by adding these distances:
[tex]\textsf{Perimeter}=CD+DE+EC\\\\\textsf{Perimeter}=\sqrt{10}+4+3\sqrt{2}\\\\\textsf{Perimeter}=11.4049183472...\\\\\textsf{Perimeter}\approx 11.4\; \sf (nearest\;tenth)[/tex]
So, the perimeter of triangle CDE is 11.4 units, rounded to the nearest tenth.
