Question

for the points p and q do the following. (a) find the distance d(p,q). (b) find the coordinates of the mid - point m of the segment pq. p(3\sqrt{2},5\sqrt{3}),q(\sqrt{2},-\sqrt{3}). (a) the distance d(p,q) is . (simplify your answer. type an exact answer, using radicals as needed.) (b) the mid - point of the segment pq is m . (simplify your answer. type an ordered pair. type an exact answer for each coordinate, using radicals as needed.)

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, $x_1 = 3\sqrt{2}$, $y_1=5\sqrt{3}$, $x_2=\sqrt{2}$, $y_2 =-\sqrt{3}$.

Step2: Calculate differences

$x_2 - x_1=\sqrt{2}-3\sqrt{2}=- 2\sqrt{2}$, $y_2 - y_1=-\sqrt{3}-5\sqrt{3}=-6\sqrt{3}$.

Step3: Square the differences

$(x_2 - x_1)^2=(-2\sqrt{2})^2 = 8$, $(y_2 - y_1)^2=(-6\sqrt{3})^2=108$.

Step4: Sum and square - root

$d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=\sqrt{8 + 108}=\sqrt{116}=2\sqrt{29}$.

Step5: Recall mid - point formula

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

Step6: Calculate mid - point coordinates

$\frac{x_1 + x_2}{2}=\frac{3\sqrt{2}+\sqrt{2}}{2}=\frac{4\sqrt{2}}{2}=2\sqrt{2}$, $\frac{y_1 + y_2}{2}=\frac{5\sqrt{3}-\sqrt{3}}{2}=\frac{4\sqrt{3}}{2}=2\sqrt{3}$.

Answer:

(a) $2\sqrt{29}$
(b) $(2\sqrt{2},2\sqrt{3})$