Question

find the perimeter of the triangle with the given vertices. use the distance formula and show work! 9. a(-3,0) b(0,4) c(0,0) 10. r(-2,1) s(2,-1) t(0,-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 length of side AB for first triangle

For points $A(-3,0)$ and $B(0,4)$, we have $x_1=-3,y_1 = 0,x_2=0,y_2 = 4$. Then $AB=\sqrt{(0 + 3)^2+(4 - 0)^2}=\sqrt{9 + 16}=\sqrt{25}=5$.

Step3: Calculate length of side BC for first triangle

For points $B(0,4)$ and $C(0,0)$, we have $x_1=0,y_1 = 4,x_2=0,y_2 = 0$. Then $BC=\sqrt{(0 - 0)^2+(0 - 4)^2}=\sqrt{0+16}=4$.

Step4: Calculate length of side AC for first triangle

For points $A(-3,0)$ and $C(0,0)$, we have $x_1=-3,y_1 = 0,x_2=0,y_2 = 0$. Then $AC=\sqrt{(0 + 3)^2+(0 - 0)^2}=\sqrt{9+0}=3$.

Step5: Calculate perimeter of first triangle

$P_{ABC}=AB + BC+AC=5 + 4+3=12$.

Step6: Calculate length of side RS for second triangle

For points $R(-2,1)$ and $S(2,-1)$, we have $x_1=-2,y_1 = 1,x_2=2,y_2=-1$. Then $RS=\sqrt{(2 + 2)^2+(-1 - 1)^2}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}$.

Step7: Calculate length of side ST for second triangle

For points $S(2,-1)$ and $T(0,-4)$, we have $x_1=2,y_1=-1,x_2=0,y_2=-4$. Then $ST=\sqrt{(0 - 2)^2+(-4 + 1)^2}=\sqrt{4 + 9}=\sqrt{13}$.

Step8: Calculate length of side RT for second triangle

For points $R(-2,1)$ and $T(0,-4)$, we have $x_1=-2,y_1 = 1,x_2=0,y_2=-4$. Then $RT=\sqrt{(0 + 2)^2+(-4 - 1)^2}=\sqrt{4 + 25}=\sqrt{29}$.

Step9: Calculate perimeter of second triangle

$P_{RST}=RS+ST + RT=2\sqrt{5}+\sqrt{13}+\sqrt{29}$.

Answer:

The perimeter of the triangle with vertices $A(-3,0),B(0,4),C(0,0)$ is $12$. The perimeter of the triangle with vertices $R(-2,1),S(2,-1),T(0,-4)$ is $2\sqrt{5}+\sqrt{13}+\sqrt{29}$.