Question

find the perimeter of the triangle whose vertices are (-4,3), (-4,1), and (-3, - 4). write the exact answer. do not round.

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 $(-4,3)$ and $(-4,1)$

Let $(x_1,y_1)=(-4,3)$ and $(x_2,y_2)=(-4,1)$. Then $d_1=\sqrt{(-4 + 4)^2+(1 - 3)^2}=\sqrt{0+( - 2)^2}=\sqrt{4}=2$.

Step3: Calculate distance between $(-4,1)$ and $(-3,-4)$

Let $(x_1,y_1)=(-4,1)$ and $(x_2,y_2)=(-3,-4)$. Then $d_2=\sqrt{(-3 + 4)^2+(-4 - 1)^2}=\sqrt{1+( - 5)^2}=\sqrt{1 + 25}=\sqrt{26}$.

Step4: Calculate distance between $(-3,-4)$ and $(-4,3)$

Let $(x_1,y_1)=(-3,-4)$ and $(x_2,y_2)=(-4,3)$. Then $d_3=\sqrt{(-4 + 3)^2+(3 + 4)^2}=\sqrt{( - 1)^2+7^2}=\sqrt{1 + 49}=\sqrt{50}=5\sqrt{2}$.

Step5: Calculate perimeter

The perimeter $P=d_1 + d_2 + d_3=2+\sqrt{26}+5\sqrt{2}$.

Answer:

$2+\sqrt{26}+5\sqrt{2}$