Answer
(a) $d=\sqrt{133}$
(b) $midpoint\displaystyle =(2\sqrt{2}, \frac{3\sqrt{5}}{2})$
Work Step by Step
We are given the points:
$P(3\sqrt{2},4\sqrt{5})$ , $Q(\sqrt{2},-\sqrt{5})$
(a) To find the distance between the two points, we use the distance formula:
$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
$=\sqrt{(\sqrt{2}-3\sqrt{2})^{2}+(-\sqrt{5}-4\sqrt{5})^{2}}$
$=\sqrt{(-2\sqrt{2})^{2}+(-5\sqrt{5})^{2}}$
$=\sqrt{8+125}$
$=\sqrt{133}$
(b) To find the midpoint, we use the midpoint formula:
$\displaystyle Midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$
$\displaystyle =(\frac{3\sqrt{2}+\sqrt{2}}{2},\frac{4\sqrt{5}+-\sqrt{5}}{2})$
$\displaystyle =(\frac{4\sqrt{2}}{2},\frac{3\sqrt{5}}{2})$
$\displaystyle =(2\sqrt{2}, \frac{3\sqrt{5}}{2})$