The equation of line passing through points $(5,0)$ and $(0,3)$ is
$(y-0)=\frac{(3-0)}{0-5}(x-5)$
$\Rightarrow y=\frac{3}{-5}(x-5)$
$\Rightarrow -5 y=3 x-15$
$\Rightarrow 3 x+5 y-15-0$
$\therefore $ The length of perpendicular from the point $(4,4)$ to the above line is
$d =\frac{(3 \times 4+5 \times 4-15)}{\sqrt{3^{2}+5^{2}}}$
$\equiv\frac{12+20-15}{\sqrt{9+25}}$
$=\frac{17}{\sqrt{34}}=\sqrt{\frac{17}{2}}$