Question

find the perimeter of the triangle whose vertices are (2, - 9), (- 1, 0), and (- 3, - 8). 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 side 1 length

Let $(x_1,y_1)=(2,-9)$ and $(x_2,y_2)=(-1,0)$. Then $d_1=\sqrt{(-1 - 2)^2+(0 + 9)^2}=\sqrt{(-3)^2+9^2}=\sqrt{9 + 81}=\sqrt{90}=3\sqrt{10}$.

Step3: Calculate side 2 length

Let $(x_1,y_1)=(-1,0)$ and $(x_2,y_2)=(-3,-8)$. Then $d_2=\sqrt{(-3 + 1)^2+(-8 - 0)^2}=\sqrt{(-2)^2+(-8)^2}=\sqrt{4 + 64}=\sqrt{68}=2\sqrt{17}$.

Step4: Calculate side 3 length

Let $(x_1,y_1)=(-3,-8)$ and $(x_2,y_2)=(2,-9)$. Then $d_3=\sqrt{(2 + 3)^2+(-9 + 8)^2}=\sqrt{5^2+(-1)^2}=\sqrt{25 + 1}=\sqrt{26}$.

Step5: Calculate perimeter

The perimeter $P=d_1 + d_2 + d_3=3\sqrt{10}+2\sqrt{17}+\sqrt{26}$.

Answer:

$3\sqrt{10}+2\sqrt{17}+\sqrt{26}$