Construct a Perpendicular Line Through a Point That Does Not Lie on Any Line

To construct a perpendicular line through a point that is not on an existing line, you can follow these steps using a compass and straightedge:

1. Identify the Point: Start by marking the point, let’s call it point P, on your paper.
2. Draw an Arbitrary Line: Using the straightedge, draw a horizontal line that runs across your paper. This line will not pass through point P.
3. Set the Compass: Place the compass point on point P and draw a circle that intersects the arbitrary line you drew. Ensure that the radius is long enough to cross the line.
4. Label the Intersections: Label the two points where the circle intersects the line as points A and B.
5. Construct the Perpendicular Bisector: Now, without changing the compass width, place the compass on point A and draw an arc above and below the line. Repeat this process with the compass centered on point B, so the arcs intersect.
6. Mark the Intersections: Label the points where the arcs intersect as points C and D.
7. Draw the Perpendicular Line: Use the straightedge to draw a line that connects points C and D. This line is the perpendicular line you wanted to construct through point P.

In conclusion, you have successfully constructed a perpendicular line through a point not on any existing line by creating a circle, finding intersection points, and drawing a perpendicular bisector. This method ensures accuracy and is widely used in geometric constructions.