Perpendicular from point to line

Z

Zongear

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

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Zongear said:
Here is the solution that I think seems valid, and the associated mark scheme.
Click to expand...
Your work is excellent. It does exactly what was asked for, and very efficiently, by reflecting P across the line.

It's possible that your construction is not taught as often as the other because it is a little harder to prove. If you were challenged, could you prove that your construction necessarily produces a perpendicular line? That might help convince them to accept it.

I don't know the meaning of "M1, M1dep, A1". But if this literally means that they only accept that one method, then they deserve no marks. I've heard of education systems that expect students to do exactly what they teach and no more, but this is ridiculous. Students need to learn how to think for themselves, and have their creativity celebrated. Please accept my congratulations.
 
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