Question

find the distance between (8,4) and (9,-2) *
your answer

find the midpoint between (8,4) and (9,-2) *
your answer

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here $x_1 = 8$, $y_1=4$, $x_2 = 9$, $y_2=-2$.

Step2: Calculate differences

$x_2 - x_1=9 - 8=1$ and $y_2 - y_1=-2 - 4=-6$.

Step3: Square differences and sum

$(x_2 - x_1)^2+(y_2 - y_1)^2=1^2+(-6)^2=1 + 36=37$.

Step4: Calculate distance

$d=\sqrt{37}$.

Step5: Recall mid - point formula

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.

Step6: Calculate mid - point coordinates

$\frac{x_1 + x_2}{2}=\frac{8 + 9}{2}=\frac{17}{2}=8.5$ and $\frac{y_1 + y_2}{2}=\frac{4+( - 2)}{2}=\frac{2}{2}=1$.

Answer:

Distance: $\sqrt{37}$
Mid - point: $(8.5,1)$