Construction: Perpendicular Line from a Point on the Line

How to construct a line that is perpendicular to the given line and that passes through a point on the line: by using a compass and a straightedge (ruler).

Example

Given

Let's see how to construct the line
that is perpendicular to the given line
and that passes through
the given point on the line.

Construction

Place the point of the compass
at the given point.

Draw two arcs
that intersect with the given line.

Place the point of the compass
at the left intersecting point
of the arc and the given line.
(blue point)

Draw two arcs
on the top and bottom of the given point.
(blue arcs)

Without changing the radius of the compass,
place the point of the compass
at the other intersecting point.

Draw two arcs (blue arcs)
that intersect the previous arcs.

Draw a line that passes through
the intersecting points
of the top arcs and the bottom arcs.

The line should pass through
the given point on the line.

Then the blue line is the line
that is perpendicular to the given line
and that passes through
the given point on the line.