Question

what is the length of rs with r(-3, 2) and s(5, 4)? *
your answer
what is the length of fg with f(-4, 3) and g(2, 5)? *
your answer
what is the length of ny with n(1, -3) and y(5, 5)? *

Explanation:

Step1: Identify the 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 the length of RS

For points $R(-3,2)$ and $S(5,4)$, let $(x_1,y_1)=(-3,2)$ and $(x_2,y_2)=(5,4)$.
$d_{RS}=\sqrt{(5 - (-3))^2+(4 - 2)^2}=\sqrt{(5 + 3)^2+(4 - 2)^2}=\sqrt{8^2+2^2}=\sqrt{64 + 4}=\sqrt{68}=2\sqrt{17}$.

Step3: Calculate the length of FG

For points $F(-4,3)$ and $G(2,5)$, let $(x_1,y_1)=(-4,3)$ and $(x_2,y_2)=(2,5)$.
$d_{FG}=\sqrt{(2-(-4))^2+(5 - 3)^2}=\sqrt{(2 + 4)^2+(5 - 3)^2}=\sqrt{6^2+2^2}=\sqrt{36+4}=\sqrt{40}=2\sqrt{10}$.

Step4: Calculate the length of NY

For points $N(1,-3)$ and $Y(5,5)$, let $(x_1,y_1)=(1,-3)$ and $(x_2,y_2)=(5,5)$.
$d_{NY}=\sqrt{(5 - 1)^2+(5-(-3))^2}=\sqrt{4^2+(5 + 3)^2}=\sqrt{16+64}=\sqrt{80}=4\sqrt{5}$.

Answer:

Length of RS: $2\sqrt{17}$
Length of FG: $2\sqrt{10}$
Length of NY: $4\sqrt{5}$