Answer
$\color{blue}{\bf{\text{(a) }\sqrt{65}}}$
$\color{blue}{\bf{\text{(b) }(\frac{5}{2},1)}}$
$\color{blue}{\bf\text{(c) }{y=8x-19}}$
Work Step by Step
$\bf\text{(a)}$ distance between points $P(3,5)$ and $Q(2,-3)$
Use the distance formula:
$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
$d(P,Q)=\sqrt{(3-2)^2+(5-(-3))^2}$
$d(P,Q)=\sqrt{1^2+8^2}$
$d(P,Q)=\sqrt{1+64}$
$d(P,Q)=\color{blue}{\bf{\sqrt{65}}}$
$\bf\text{(b)}$ midpoint between points $P(3,5)$ and $Q(2,-3)$
Use the midpoint formula:
$$m=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$
$m=(\frac{3+2}{2},\frac{5+(-3)}{2})$
$m=(\frac{5}{2},\frac{2}{2})$
$m=\color{blue}{\bf{(\frac{5}{2},1)}}$
$\bf\text{(c)}$ equation for the line that passes through the points $P(3,5)$ and $Q(2,-3)$ in slope intercept form
Use point slope form:
$$y-y_1=m(x-x_1)$$
where $m$ = slope = $\frac{\Delta{y}}{\Delta{x}}$ =$\frac{y_1-y_2}{x_1-x_2}$
$y-5=\frac{5-(-3)}{3-2}(x-3)$
$y-5={\frac{8}{1}}(x-3)$
$y-5=8(x-3)$
$y-5=8x-24$
$\color{blue}{\bf
{y=8x-19}}$
