Construct the line through P that is perpendicular to line g.

Hint: Use the perpendicular bisector between two points on g that are equally distant to  P (How will you construct the two points?)

The solution is here.

Is it possible to solve this problem, if P is a point on g?

When using C.a.R., you can do this entire construction easily by making use of the "Perpendicular line (2)" tool:

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Just click on the tool, then click on the line, then click on the point (the instructions for using the tool appear under the construction window when you click the tool's icon)

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