Question

a line segment is drawn on a coordinate plane with the endpoints r(-4, -2) and s(4, 4). point t is the midpoint of $overline{rs}$. what is the length of $overline{rs}$? what is the length of $overline{st}$? what are the coordinates of point t?

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $R(-4,-2)$ and $S(4,4)$, we have $x_1=-4,y_1 = - 2,x_2=4,y_2 = 4$.

$$d_{RS}=\sqrt{(4 - (-4))^2+(4-(-2))^2}=\sqrt{(4 + 4)^2+(4 + 2)^2}=\sqrt{8^2+6^2}=\sqrt{64 + 36}=\sqrt{100}=10$$

Step2: Find mid - point formula

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. For points $R(-4,-2)$ and $S(4,4)$, the coordinates of $T$ are $(\frac{-4 + 4}{2},\frac{-2+4}{2})=(0,1)$.

Step3: Find length of $\overline{ST}$

Using the distance formula for points $S(4,4)$ and $T(0,1)$, we have $x_1 = 4,y_1=4,x_2=0,y_2 = 1$. Then $d_{ST}=\sqrt{(0 - 4)^2+(1 - 4)^2}=\sqrt{(-4)^2+(-3)^2}=\sqrt{16 + 9}=\sqrt{25}=5$

Answer:

Length of $\overline{RS}$: $10$
Length of $\overline{ST}$: $5$
Coordinates of point $T$: $(0,1)$