Question

find the perimeter of the polygon with the given vertices l(1, 4), n(2, 0), m(4, 0), p( - 1, - 2)

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 $LP$

For points $L(1,4)$ and $P(1, - 2)$, $x_1 = 1,y_1 = 4,x_2 = 1,y_2=-2$. Then $d_{LP}=\sqrt{(1 - 1)^2+( - 2 - 4)^2}=\sqrt{0+( - 6)^2}=6$.

Step3: Calculate distance $PN$

For points $P(1,-2)$ and $N(2,0)$, $x_1 = 1,y_1=-2,x_2 = 2,y_2 = 0$. Then $d_{PN}=\sqrt{(2 - 1)^2+(0 + 2)^2}=\sqrt{1 + 4}=\sqrt{5}$.

Step4: Calculate distance $NM$

For points $N(2,0)$ and $M(4,0)$, $x_1 = 2,y_1 = 0,x_2 = 4,y_2 = 0$. Then $d_{NM}=\sqrt{(4 - 2)^2+(0 - 0)^2}=\sqrt{4+0}=2$.

Step5: Calculate distance $ML$

For points $M(4,0)$ and $L(1,4)$, $x_1 = 4,y_1 = 0,x_2 = 1,y_2 = 4$. Then $d_{ML}=\sqrt{(1 - 4)^2+(4 - 0)^2}=\sqrt{( - 3)^2+16}=\sqrt{9 + 16}=5$.

Step6: Calculate perimeter

The perimeter $P=d_{LP}+d_{PN}+d_{NM}+d_{ML}=6+\sqrt{5}+2 + 5=13+\sqrt{5}$.

Answer:

$13+\sqrt{5}$