Question

which triangle has the greatest perimeter?
△efg with vertices e(-7, 1), f(1, -1), and g(-7, -5)
△jkl with vertices j(-1, 2), k(6, 6), and l(7, 2)
△nop with vertices n(-2, 8), o(4, 9), and p(4, 4)

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 - lengths and perimeter of $\triangle EFG$

• $EF=\sqrt{(1+7)^2+(-1 - 1)^2}=\sqrt{64 + 4}=\sqrt{68}=2\sqrt{17}$
• $FG=\sqrt{(-7 - 1)^2+(-5 + 1)^2}=\sqrt{64+16}=\sqrt{80}=4\sqrt{5}$
• $EG=\sqrt{(-7 + 7)^2+(-5 - 1)^2}=\sqrt{36}=6$
• Perimeter of $\triangle EFG=2\sqrt{17}+4\sqrt{5}+6\approx2\times4.123+4\times2.236 + 6=8.246+8.944+6=23.19$

Step3: Calculate side - lengths and perimeter of $\triangle JKL$

• $JK=\sqrt{(6 + 1)^2+(6 - 2)^2}=\sqrt{49 + 16}=\sqrt{65}$
• $KL=\sqrt{(7 - 6)^2+(2 - 6)^2}=\sqrt{1 + 16}=\sqrt{17}$
• $JL=\sqrt{(7 + 1)^2+(2 - 2)^2}=8$
• Perimeter of $\triangle JKL=\sqrt{65}+\sqrt{17}+8\approx8.062+4.123+8=20.185$

Step4: Calculate side - lengths and perimeter of $\triangle NOP$

• $NO=\sqrt{(4 + 2)^2+(9 - 8)^2}=\sqrt{36+1}=\sqrt{37}$
• $OP=\sqrt{(4 - 4)^2+(4 - 9)^2}=5$
• $NP=\sqrt{(4 + 2)^2+(4 - 8)^2}=\sqrt{36 + 16}=\sqrt{52}=2\sqrt{13}$
• Perimeter of $\triangle NOP=\sqrt{37}+5+2\sqrt{13}\approx6.083+5+2\times3.606=6.083+5+7.212=18.295$

Answer:

$\triangle EFG$ has the greatest perimeter.