Explanation

(c)
$\therefore Median\ vector\ through\ \overrightarrow{\mathbf{A}} = \frac{1}{2}\left( \overrightarrow{\mathbf{AB}} + \overrightarrow{\mathbf{AC}} \right)$
$= \frac{1}{2}\left\lbrack \left( 3\widehat{\dot{\mathbf{i}}} + 5\widehat{\dot{\mathbf{j}}} + 4\widehat{\mathbf{k}} \right) + \left( 5\widehat{\dot{\mathbf{i}}} - 5\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}} \right) \right\rbrack$
$= 4\widehat{\dot{\mathbf{i}}} + 3\widehat{\mathbf{k}}$
$\therefore$ Length of the median $= \sqrt{4^{2} + 3^{2}} = 5$ units