Solution

To construct a tangent to a circle with center P and radius 3.2 cm at any point M on it, follow these steps:

1. Draw a circle with center P and radius 3.2 cm.

2. Choose any point M on the circumference of the circle.

3. Draw a line segment PM to connect point P with point M.

4. Draw a line perpendicular to line segment PM at point M. To do this, use a compass to draw an arc centered at M that intersects the circle at points X and Y. Then draw a line through point M that is perpendicular to line segment XY.

5. Draw a line segment PT that passes through point P and is perpendicular to line segment PM. To do this, use a compass to draw an arc centered at P that intersects the circle at points Z and W. Then draw a line through point P that passes through the midpoint of line segment ZW.

6. Line segment PT is the required tangent to the circle at point M.

Note that the tangent is unique, and it is perpendicular to the radius of the circle that passes through point M.