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. scratch area (not part of answer)

Explanation:

Step1: Recall the distance formula

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

Step2: Find the length of one - side

Let's find the distance between $(1,4)$ and $(5, - 5)$.
$x_1 = 1,y_1 = 4,x_2 = 5,y_2=-5$.
$d_1=\sqrt{(5 - 1)^2+(-5 - 4)^2}=\sqrt{4^2+(-9)^2}=\sqrt{16 + 81}=\sqrt{97}$.

Step3: Find the length of an adjacent - side

Let's find the distance between $(1,4)$ and $(-6,4)$.
$x_1 = 1,y_1 = 4,x_2=-6,y_2 = 4$.
$d_2=\sqrt{(-6 - 1)^2+(4 - 4)^2}=\sqrt{(-7)^2+0^2}=7$.

Step4: Calculate the perimeter of the parallelogram

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

Answer:

$2\sqrt{97}+14$