Question

consider the following: (3, 9), (9, 3) (a) plot the points. (b) find the distance between the points. (c) find the mid - point of the line segment joining the points.

Explanation:

Step1: Recall distance formula

The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $(x_1,y_1)=(3,9)$ and $(x_2,y_2)=(9,3)$.

Step2: Substitute values into formula

$d=\sqrt{(9 - 3)^2+(3 - 9)^2}=\sqrt{6^2+( - 6)^2}=\sqrt{36 + 36}=\sqrt{72}=6\sqrt{2}$

Step3: Recall mid - point formula

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

Step4: Substitute values into mid - point formula

$M=(\frac{3 + 9}{2},\frac{9+3}{2})=(6,6)$

Answer:

(b) $6\sqrt{2}$
(c) $(6,6)$

(Note: Plotting the points in part (a) is a visual task that cannot be fully represented in this text - based format. You would go 3 units to the right and 9 units up from the origin to plot the point $(3,9)$ and 9 units to the right and 3 units up from the origin to plot the point $(9,3)$ on the given graph.)