Answer
$\color{blue}{\bf\text{(a) }2}$
$\color{blue}{\bf\text{(b) }(5,0)}$
$\color{blue}{\bf\text{(c) } x=5 }$
Work Step by Step
$\bf{(a)}$ distance between points $\bf{P(5,-1)}$ and $\bf{Q(5,1)}$
Use the distance formula:
$$d=\sqrt{(x_2−x_1)^2+(y_2−y_1)^2}$$
$d=\sqrt{(5-5)^2+(-1-1)^2}$
$d=\sqrt{0^2+(-2)^2}$
$d=\sqrt{4}$
$d=\color{blue}{\bf{2}}$
$\bf{(b)}$ midpoint between points $\bf{P(5,-1)}$ and $\bf{Q(5,1)}$
Use the midpoint formula:
$$m=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$
$m=(\frac{5+5}{2},\frac{-1+1}{2})$
$m=(\frac{10}{2},\frac{0}{2})$
$m=\color{blue}{\bf{(5,0)}}$
$\bf{(c)}$ equation for the line that passes through the points $\bf{P(5,-1)}$ and $\bf{Q(5,1)}$
in slope intercept form
Use point slope form, $y−y_1=m(x−x_1)$
where $\bf{m}$ = slope = $\frac{\Delta{y}}{\Delta{x}} =\frac{y_1−y_2}{x_1−x_2}$
$$y−y_1=\frac{y_1−y_2}{x_1−x_2}(x−x_1)$$
We already see that the $x$ value is the same for both points so the slope is:
${\frac{-1-1}{5-5}}={\frac{0}{\bf0}}$
and since division by zero is undefined, the formula of the vertical line is:
$\color{blue}{\bf{x=5}}$
$\color{blue}{\bf\text{(a) }2}$
$\color{blue}{\bf\text{(b) }(5,0)}$
$\color{blue}{\bf\text{(c) } x=5 }$
