Hello there. To solve this question, we have to remember some properties about coordinate geometry and the pythagorean theorem.
Given the following triangle in the xy plane:
To determine the length of the side AD, consider the following auxiliary triangle:
Notice it is a right triangle, that has hypotenuse with the same length as AD.
The legs of this triangle are given by the distance between each coordinate of the points A and D.
We find, by inspection, that the coordinates of A are (-5, 5) and D are (8, 3)
This means that the horizontal distance between A and D x coordinates are 13 and the height is 2.
The length is then given by the pythagorean theorem:
[tex]\begin{gathered} L_{AD}^2=13^2+2^2=169+4=173 \\ \\ L_{AD}=\sqrt{173}\approx13.15 \\ \end{gathered}[/tex]
To determine AB, consider the next triangle:
B has coordinates (1, 1), so its horizontal distance to A is 6 and its vertical distance is 4.
The hypotenuse length is the same as the side AB, hence
[tex]\begin{gathered} L_{AB}^2=6^2+4^2=36+16=52 \\ \\ L_{AB}=\sqrt{52}\approx7.21 \end{gathered}[/tex]
Finally, the side BD can also be calculated in the same manner:
The horizontal distance between B and D is 7 and the vertical distance is 2.
The length of BD is then given by
[tex]\begin{gathered} L_{BD}^2=7^2+2^2=49+4=53 \\ \\ L_{BD}=\sqrt{53}\approx7.28 \\ \end{gathered}[/tex]
The perimeter is then given by the sum of the length of each side of the triangle:
[tex]p\approx13.15+7.21+7.28=27.64[/tex]
These are the answers to this question.
