In the given figure, ABCD is a square whose side is 4 cm. P is a point on the side AD. What is the minimum value (in cm) of BP + CP?

Let P is the mid point of AD
$$\eqalign{ & BP = \sqrt {A{P^2} + A{B^2}} \cr & = \sqrt {A{P^2} + 16} \cr & = \sqrt {{2^2} + 16} \cr & = \sqrt {20} \cr & {\text{Similarly }}CP = \sqrt {20} \cr & BP + CP = \sqrt {20} + \sqrt {20} = 4\sqrt 5 \cr} $$