Question

find the perimeter of the polygon with the given vertices. round your answer to the nearest hundredth. the perimeter is about 14.05 x units.

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: Calculate distance between $(-1,-2)$ and $(1,4)$

Let $(x_1,y_1)=(-1,-2)$ and $(x_2,y_2)=(1,4)$. Then $d_1=\sqrt{(1 - (-1))^2+(4 - (-2))^2}=\sqrt{(2)^2+(6)^2}=\sqrt{4 + 36}=\sqrt{40}=2\sqrt{10}\approx 6.32$.

Step3: Calculate distance between $(1,4)$ and $(4,0)$

Let $(x_1,y_1)=(1,4)$ and $(x_2,y_2)=(4,0)$. Then $d_2=\sqrt{(4 - 1)^2+(0 - 4)^2}=\sqrt{(3)^2+(-4)^2}=\sqrt{9 + 16}=\sqrt{25}=5$.

Step4: Calculate distance between $(4,0)$ and $(2,0)$

Let $(x_1,y_1)=(4,0)$ and $(x_2,y_2)=(2,0)$. Then $d_3=\sqrt{(2 - 4)^2+(0 - 0)^2}=\sqrt{(-2)^2+0^2}=2$.

Step5: Calculate distance between $(2,0)$ and $(-1,-2)$

Let $(x_1,y_1)=(2,0)$ and $(x_2,y_2)=(-1,-2)$. Then $d_4=\sqrt{(-1 - 2)^2+(-2 - 0)^2}=\sqrt{(-3)^2+(-2)^2}=\sqrt{9+4}=\sqrt{13}\approx 3.61$.

Step6: Calculate perimeter

$P=d_1 + d_2 + d_3 + d_4\approx6.32+5 + 2+3.61 = 16.93$.

Answer:

$16.93$