Question

find the perimeter of the parallelogram with these vertices. (1, 4), (-6, 4), (5, -5), (-2, -5) give an exact answer (not a decimal approximation). simplify your answer as much as possible.

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Find length of one - side

Let's find the distance between $(1,4)$ and $(-6,4)$. Here $x_1 = 1,y_1 = 4,x_2=-6,y_2 = 4$. Then $d_1=\sqrt{(-6 - 1)^2+(4 - 4)^2}=\sqrt{(-7)^2+0^2}=7$.

Step3: Find length of another - side

Find the distance between $(1,4)$ and $(5,-5)$. Here $x_1 = 1,y_1 = 4,x_2 = 5,y_2=-5$. Then $d_2=\sqrt{(5 - 1)^2+(-5 - 4)^2}=\sqrt{4^2+(-9)^2}=\sqrt{16 + 81}=\sqrt{97}$.

Step4: Calculate perimeter of parallelogram

In a parallelogram, opposite sides are equal. The perimeter $P$ of a parallelogram with adjacent - side lengths $a$ and $b$ is $P = 2(a + b)$. Here $a = 7$ and $b=\sqrt{97}$, so $P=2(7+\sqrt{97})=14 + 2\sqrt{97}$.

Answer:

$14 + 2\sqrt{97}$