Question

find the perimeter of the parallelogram with these vertices. (3,5),(-4,2),(3,1),(-4,-2)

Explanation:

Step1: Calculate the length of one side

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Take \((3,5)\) and \((3,1)\):
\(d_1=\sqrt{(3 - 3)^2+(1 - 5)^2}=\sqrt{0+(- 4)^2}=4\)

Step2: Calculate the length of another side

Take \((3,5)\) and \((-4,2)\):
\(d_2=\sqrt{(-4 - 3)^2+(2 - 5)^2}=\sqrt{(-7)^2+(-3)^2}=\sqrt{49 + 9}=\sqrt{58}\)

Step3: Calculate the perimeter

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

Answer:

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