Question

use the given coordinates to compute the perimeter of the parallelogram.

Explanation:

Step1: Calculate the length of \(AB\)

Since \(A(-5,-4)\) and \(B(2,-4)\), using the distance formula for points with the same \(y\) - coordinate \(d=\vert x_2 - x_1\vert\).
\(AB=\vert2-(-5)\vert=\vert2 + 5\vert = 7\)

Step2: Calculate the length of \(BC\)

Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(B(2,-4)\) and \(C(-3,2)\)
\(x_1 = 2,y_1=-4,x_2=-3,y_2 = 2\)
\(BC=\sqrt{(-3 - 2)^2+(2-(-4))^2}=\sqrt{(-5)^2+(6)^2}=\sqrt{25 + 36}=\sqrt{61}\)

Step3: Use the property of parallelogram

In a parallelogram \(AB = CD\) and \(BC=AD\). The perimeter \(P = 2(AB + BC)\)
Substitute \(AB = 7\) and \(BC=\sqrt{61}\) into the formula
\(P=2(7+\sqrt{61})=14 + 2\sqrt{61}\approx14+2\times7.81=14 + 15.62=29.62\)

Answer:

The perimeter of the parallelogram is \(14 + 2\sqrt{61}\approx29.62\)