the coodrinates of JKLM is,
J(1,3) , K(3,1), L(-1,-3) and M(-3,-1)
the length JK is ,
![\begin{gathered} JK=\sqrt[]{(1-3)^2+(3-1)^2} \\ =\sqrt[]{4+4} \\ =\sqrt[]{8} \\ =2\sqrt[]{2} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/high-school/qrrh6n6i5i0mxp77qlj4antltrrvixn6w8.png)
length KL is,
![\begin{gathered} KL=\sqrt[]{(-3-1)^2+(-1-3)^2} \\ =\sqrt[]{16+16} \\ =\sqrt[]{32} \\ =4\sqrt[]{2} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/high-school/lr64cqc342wekzjg528wop5dt2170jmmnz.png)
Length LM is,
![LM=\sqrt[]{(-3-(-1))^2+(-1-(-3))^2}](https://img.qammunity.org/2023/formulas/mathematics/high-school/nncikvgdfbmnju7856eowcjy15bvnndobo.png)
![\begin{gathered} LM=\sqrt[]{4+4} \\ LM=2\sqrt[]{2} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/high-school/vuonx4ksxj6dtrrms1k6yfho9oog66ipe9.png)
length MJ
![\sqrt[]{(-1-3)^2+(-3-1)^2}](https://img.qammunity.org/2023/formulas/mathematics/high-school/34lvluayy38pqmdq65nineokkhbhic1qvo.png)
![\begin{gathered} MJ=\sqrt[]{16+16} \\ MJ=4\sqrt[]{2} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/high-school/yn73p0jfeognee8gll2hi67n6ifgkvmu1o.png)
so, the perimeter = sum of sides,
= JK + KL + LM + MJ
= 2 root 2 + 4root2 + 2root2 + 4root2
= 12 root 2
thus, the perimeter is 12 root 2