Question

find the distance between the two points and the midpoint of the line segment joining them. (0,2) and (-3,3)

Explanation:

Step1: Recall distance formula

The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $(x_1,y_1)=(0,2)$ and $(x_2,y_2)=(-3,3)$.
$d=\sqrt{(-3 - 0)^2+(3 - 2)^2}$

Step2: Simplify distance - formula expression

$d=\sqrt{(-3)^2+1^2}=\sqrt{9 + 1}=\sqrt{10}$

Step3: Recall mid - point formula

The mid - point $M$ of the line segment joining $(x_1,y_1)$ and $(x_2,y_2)$ is $M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.
$M=(\frac{0+( - 3)}{2},\frac{2 + 3}{2})$

Step4: Calculate mid - point coordinates

$M=(-\frac{3}{2},\frac{5}{2})$

Answer:

Distance: $\sqrt{10}$; Mid - point: $(-\frac{3}{2},\frac{5}{2})$