Perpendicular from point to line

[Zongear]

I have a question asking to construct a perpendicular from a point to a given line. The point is not on the line and is not equidistant from both ends.

The mark schemes for this question only awards marks for the following method:
Draw arcs from the point that cross the line in two places
Draw arcs from those crossing points
Connect where the arcs intersect

Is it not possible to do the following (seemingly easier) steps instead?:
Draw arcs from each end of the line, ensuring each arc passes through the point.
Connect where the arcs intersect with each other

The second seems more intuitive - am I missing something?

 

[Dr.Peterson]

Can you show the actual question and the mark scheme? That may make a difference.

Certainly, given a line segment (that is, its end points) and a point not on the line, your construction works, and is easy to prove. I find it mentioned here, for instance.

Theirs is standard when you are not given two particular points on the line, and is based on the standard construction of a perpendicular bisector. It also has the slight advantage of needed only one setting of the compass. Given that they have taught this method, it is natural that that is what they expect; but it would be wrong to require it, if that's what they do.

 

[Zongear]

Here is the solution that I think seems valid, and the associated mark scheme.