Answer
$\color{blue}{\bf\text{(a) }\sqrt{10}}$
$\color{blue}{\bf\text{(b) }(\frac{3\sqrt{2}}{2},2\sqrt{2})}$
$\color{blue}{\bf\text{(c) }y=-2x+5\sqrt{2}}$
Work Step by Step
$\bf{(a)}$ distance between points
$\bf{P(2\sqrt{2},\sqrt{2})}$ and
$\bf{Q(\sqrt{2},3\sqrt{2})}$
Use the distance formula:
$$d=\sqrt{(x_2−x_1)^2+(y_2−y_1)^2}$$
$d=\sqrt{(2\sqrt{2}-\sqrt{2})^2+(\sqrt{2}-3\sqrt{2})^2}$
$d=\sqrt{((2-1)\sqrt{2})^2+((-3+1)\sqrt{2})^2}$
$d=\sqrt{(\sqrt{2})^2+(-2\sqrt{2})^2}$
$d=\sqrt{2+4(2)}$
$d=\sqrt{2+8}$
$d=\bf\color{blue}{\sqrt{10}}$
$\bf{(b)}$ midpoint between points
$\bf{P(2\sqrt{2},\sqrt{2})}$ and
$\bf{Q(\sqrt{2},3\sqrt{2})}$
Use the midpoint formula:
$$m=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$
$m=(\frac{2\sqrt{2}+\sqrt{2}}{2},\frac{\sqrt{2}+3\sqrt{2}}{2})$
$m=(\frac{(2+1)\sqrt{2}}{2},\frac{(3+1)\sqrt{2}}{2})$
$m=(\frac{3\sqrt{2}}{2},\frac{4\sqrt{2}}{2})$
$m=\color{blue}{\bf{(\frac{3\sqrt{2}}{2},2\sqrt{2})}}$
$\bf{(c)}$ equation for the line that passes through the points
$\bf{P( 2\sqrt{2} , \sqrt{2} )}$ and
$\bf{Q( \sqrt{2} , 3\sqrt{2} )}$
in slope intercept form
Use point slope form, $y−y_1=m(x−x_1)$
where $\bf{m}$ = slope = $\frac{\Delta{y}}{\Delta{x}} =\frac{y_1−y_2}{x_1−x_2}$
$$y−y_1=\frac{y_1−y_2}{x_1−x_2}(x−x_1)$$
$y−\sqrt{2}=\frac{\sqrt{2}-3\sqrt{2}}{2\sqrt{2}-\sqrt{2}}(x−2\sqrt{2})$
$y−\sqrt{2}=\frac{(-3+1)\sqrt{2}}{(2-1)\sqrt{2}}(x−2\sqrt{2})$
$y−\sqrt{2}=\frac{(-2)\sqrt{2}}{(1)\sqrt{2}}(x−2\sqrt{2})$
$y−\sqrt{2}=-2(x−2\sqrt{2})$
$y−\sqrt{2}=-2x+4\sqrt{2}$
$\color{blue}{\bf{y=-2x+5\sqrt{2}
}}$
$\color{blue}{\bf\text{(a) }\sqrt{10}}$
$\color{blue}{\bf\text{(b) }(\frac{3\sqrt{2}}{2},2\sqrt{2})}$
$\color{blue}{\bf\text{(c) }y=-2x+5\sqrt{2}}$
