Answer
(a) $2\sqrt (17)$
(b) $(5,2)$
Work Step by Step
Let $P=(x_{1},y_{1})=(6,-2)$ and $Q=(x_{2},y_{2})=(4,6)$
Part (a):
Finding the distance between P and Q,
$d(P,Q)=\sqrt ((x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2})$
$d(P,Q)=\sqrt ((4-6)^{2}+(6-(-2))^{2})$
$d(P,Q)=\sqrt ((-2)^{2}+(6+2)^{2})$
$d(P,Q)=\sqrt ((-2)^{2}+(8)^{2})$
$d(P,Q)=\sqrt (4+64)$
$d(P,Q)=\sqrt (68)$
$d(P,Q)=\sqrt (4\times17)$
$d(P,Q)=\sqrt (4)\sqrt (17)$
$d(P,Q)=2\sqrt (17)$
Part (b):
The midpoint formula is $(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$
Substituting the values, the formula becomes:
$(\frac{6+4}{2},\frac{-2+6}{2})=(\frac{10}{2},\frac{4}{2})=(5,2)$
Therefore, the coordinates of the midpoint are: $(5,2)$
