On the plane, the four points form the shape below
Notice that it is a parallelogram.
1) Since it is a parallelogram, AD=BC and AB=CD. Using the formula for the distance between two points on the plane,
[tex]d((x_1,y_1),(x_2,y_2))=\sqrt[]{(x_1-x_2)^2+(y_1-y_2)^2}[/tex]
Then,
[tex]\begin{gathered} AD=\sqrt[]{(2-4)^2+(3-6)^2}=\sqrt[]{4+9}=\sqrt[]{13} \\ \text{and} \\ AB=\sqrt[]{(2-7)^2+(3-3)^2}=\sqrt[]{5^2}=5 \end{gathered}[/tex]
Therefore, the dimensions of the yard are sqrt13, 5 or 3.605551..., 5
2) The perimeter is
[tex]P=2\sqrt[]{13}+2\cdot5=2\sqrt[]{13}+10\approx17.211102\ldots[/tex]
The perimeter is 17.211102 units or 2sqrt13+10
3) As stated above, it is a parallelogram
