M moves on average line of \(\Delta ABC\) which is matched from the midpoint of AB to the midpoint of BC

Use Cauchy:

P\(=\)\(\dfrac{4}{5}\left(x+y\right)+\left(\dfrac{6}{5}x+\dfrac{30}{x}\right)+\left(\dfrac{y}{5}+\dfrac{5}{y}\right)\)\(\ge\dfrac{4}{5}.10+2\sqrt{\dfrac{6}{5}x.\dfrac{30}{x}}+2\sqrt{\dfrac{y}{5}.\dfrac{5}{y}}\)

\(=8+12+2=22\)

minP=22\(\Leftrightarrow x=y=5\)