Question

find the perimeter of the parallelogram with these vertices. (3,5), (-4,2), (3,1), (-4,-2) give an exact answer (not a decimal approximation). simplify your answer as much as possible.

Explanation:

Step1: Use the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Let's find the distance between \((3,5)\) and \((3,1)\):
\(d_1=\sqrt{(3 - 3)^2+(1 - 5)^2}=\sqrt{0+( - 4)^2}=\sqrt{16} = 4\)
Let's find the distance between \((3,5)\) and \((-4,2)\):
\(d_2=\sqrt{(-4 - 3)^2+(2 - 5)^2}=\sqrt{(-7)^2+(-3)^2}=\sqrt{49 + 9}=\sqrt{58}\)

Step2: Use the perimeter formula of a parallelogram

The perimeter \(P\) of a parallelogram is \(P = 2(a + b)\), where \(a\) and \(b\) are the lengths of adjacent sides.
Here \(a = 4\) and \(b=\sqrt{58}\)
\(P=2(4+\sqrt{58})=8 + 2\sqrt{58}\)

Answer:

\(8 + 2\sqrt{58}\)