Question

place the value that best completes the sentence. a triangle has vertices at (1,3), (2, - 3) and (-1,-1). the perimeter of the triangle is approximately units. 10 14 16 20

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 $(1,3)$ and $(2,-3)$

Let $(x_1,y_1)=(1,3)$ and $(x_2,y_2)=(2,-3)$. Then $d_1=\sqrt{(2 - 1)^2+(-3 - 3)^2}=\sqrt{1+( - 6)^2}=\sqrt{1 + 36}=\sqrt{37}\approx6.08$.

Step3: Calculate distance between $(2,-3)$ and $(-1,-1)$

Let $(x_1,y_1)=(2,-3)$ and $(x_2,y_2)=(-1,-1)$. Then $d_2=\sqrt{(-1 - 2)^2+(-1+3)^2}=\sqrt{(-3)^2+2^2}=\sqrt{9 + 4}=\sqrt{13}\approx3.61$.

Step4: Calculate distance between $(-1,-1)$ and $(1,3)$

Let $(x_1,y_1)=(-1,-1)$ and $(x_2,y_2)=(1,3)$. Then $d_3=\sqrt{(1 + 1)^2+(3 + 1)^2}=\sqrt{2^2+4^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\approx4.47$.

Step5: Calculate perimeter

$P=d_1 + d_2 + d_3\approx6.08+3.61+4.47 = 14.16\approx14$.

Answer:

14