Perpendicular at the Vertex of a Ray

If the point where you want to construct a perpendicular is the vertex of a ray or endpoint of a segment, then you simply use the straight-edge to extend the ray or segment so the point is no longer the endpoint and do as before: make the point the midpoint of some segment and construct the perpendicular bisector of that segment.

Given:  Segment AB

Construct a line perpendicular to AB through point A.

Solution:

Place the straight-edge along AB (points A and B determine only one line) and extend the segment to any convenient length to point C so that A is between C and B: 

Then place the point of the compass at A and pencil at C to measure the distance AC, and mark off point D between A and B using an arc of this radius:

Finally, construct the perpendicular bisector of CD:

Another method is to make use of the fact that an angle inscribed in a semicircle is a right angle:

Here is an animation of that method.

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