To finsd the lengths in a plane, we can use the coordinates of the endpoints of the segment we want the length of.
Given the endpoitn, the distance between them is:
[tex]d=\sqrt[]{(x_1-x_2)^2+(y_1-y_2)^2}[/tex]
For the length of MN, we ned to use the andpoints M and N, which are at coordinates:
[tex]\begin{gathered} M=(-6,2) \\ N=(-3,6) \end{gathered}[/tex]
So, MN is:
[tex]\begin{gathered} MN=\sqrt[]{(-6-(-3))^2+(2-6)^2} \\ MN=\sqrt[]{(-6+3)^2+(-4)^2} \\ MN=\sqrt[]{(-3)^2+16} \\ MN=\sqrt[]{9+16} \\ MN=\sqrt[]{25} \\ MN=5 \end{gathered}[/tex]
Thus, the length of MN is 5.
For the length of MP, we have the points:
[tex]\begin{gathered} M=(-6,2) \\ P=(3,2) \end{gathered}[/tex]
Thus:
[tex]\begin{gathered} MP=\sqrt[]{(-6-3)^2+(2-2)^2} \\ MP=\sqrt[]{(-9)^2+(0)^2} \\ MP=\sqrt[]{81} \\ MP=9 \end{gathered}[/tex]
Thus, the length of MP is 9.
The perimeter of a figure is the sum of its sides, so it will be:
[tex]P=MN+MP+PO+NO[/tex]
But opposite sides of a parallelogram are congruent, so:
[tex]\begin{gathered} PO=MN=5 \\ NO=MP=9 \end{gathered}[/tex]
Thus:
[tex]\begin{gathered} P=2MN+2MP \\ P=2\cdot5+2\cdot9 \\ P=10+18 \\ P=28 \end{gathered}[/tex]
Thus, the perimeter is 28.
