Question

directions - find the distance, slope, and midpoint of the segment connecting the following points: (8,2) and (2,10)
midpoint formula: $\left(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2}\
ight)$
distance = 10
slope =
midpoint = (5,6)

Explanation:

Step1: Calculate the slope

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let $(x_1,y_1)=(8,2)$ and $(x_2,y_2)=(2,10)$. Then $m=\frac{10 - 2}{2 - 8}=\frac{8}{-6}=-\frac{4}{3}$.

Step2: Confirm the mid - point

Given the mid - point formula $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$, substituting $x_1 = 8,x_2 = 2,y_1 = 2,y_2 = 10$ gives $(\frac{8 + 2}{2},\frac{2+10}{2})=(5,6)$.

Step3: Confirm the distance

The distance formula is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Substituting $x_1 = 8,x_2 = 2,y_1 = 2,y_2 = 10$:
\[

$$\begin{align*} d&=\sqrt{(2 - 8)^2+(10 - 2)^2}\\ &=\sqrt{(-6)^2+8^2}\\ &=\sqrt{36 + 64}\\ &=\sqrt{100}\\ &=10 \end{align*}$$

\]

Answer:

Slope = $-\frac{4}{3}$
Midpoint = $(5,6)$
Distance = $10$