Question

for the points (8, - 9) and (2, - 7), (a) find the exact distance between the points. (b) find the midpoint of the line segment whose endpoints are the given points.

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 $x_1 = 8,y_1=-9,x_2 = 2,y_2=-7$.

Step2: Substitute values into formula

$d=\sqrt{(2 - 8)^2+(-7+9)^2}=\sqrt{(-6)^2+2^2}=\sqrt{36 + 4}=\sqrt{40}=2\sqrt{10}$

Step3: Recall mid - point formula

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

Step4: Substitute values into mid - point formula

$M=(\frac{8 + 2}{2},\frac{-9-7}{2})=(5,-8)$

Answer:

(a) $2\sqrt{10}$
(b) $(5,-8)$