If a point P moves so that it is equidistant from points L and M, then it must lie on the perpendicular bisector of the line segment LM. Let's say that the distance between P and LM is x.

Since P is equidistant from L and M, we can use the Pythagorean theorem to find the relationship between x and the distance between P and L (which is given as 10cm).

Let's say that the midpoint of LM is N. Then, LN = 16/2 = 8cm.

We can draw a right triangle with hypotenuse LP, one leg PN of length x, and one leg NL of length 8cm.

Using the Pythagorean theorem, we have:

\(LP^2 = PN^2 + NL^2\), so

\(10^2 = x^2 + 8^2\).

Solving for x, we get:

x = √(10^2 - 8^2) = √(100 - 64) = √36 = 6.

So, the distance of P from LM when P is 10cm from L is 6cm.