Question

find the perimeter of the triangle whose vertices are (-5, - 1), (7, 15), and (-5, 6). 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 $(-5,-1)$ and $(7,15)$

Let $(x_1,y_1)=(-5,-1)$ and $(x_2,y_2)=(7,15)$. Then $d_1=\sqrt{(7 - (-5))^2+(15 - (-1))^2}=\sqrt{(7 + 5)^2+(15 + 1)^2}=\sqrt{12^2+16^2}=\sqrt{144 + 256}=\sqrt{400}=20$.

Step3: Calculate distance between $(7,15)$ and $(-5,6)$

Let $(x_1,y_1)=(7,15)$ and $(x_2,y_2)=(-5,6)$. Then $d_2=\sqrt{(-5 - 7)^2+(6 - 15)^2}=\sqrt{(-12)^2+(-9)^2}=\sqrt{144+81}=\sqrt{225}=15$.

Step4: Calculate distance between $(-5,6)$ and $(-5,-1)$

Let $(x_1,y_1)=(-5,6)$ and $(x_2,y_2)=(-5,-1)$. Then $d_3=\sqrt{(-5 - (-5))^2+(-1 - 6)^2}=\sqrt{0^2+(-7)^2}=\sqrt{49}=7$.

Step5: Calculate perimeter

The perimeter $P$ of the triangle is $P=d_1 + d_2 + d_3=20+15 + 7=42$.

Answer:

42