Locus of all points -- Imprecise question?

[paulb203]

TL;DR Summary: When asked to show the LOCUS OF ALL POINTS how precise should the questioner be?

I was asked to draw THE LOCUS OF ALL POINTS 3cm from a line.

I measured 3cm above and below the line, and drew two parallel lines there. Then drew two semi-circles with radii 3cm out from either end of the line to complete the locus (a shape which Google tells me is a stadium).

But is the question imprecise?

Regards the lines above and below the line; shouldn’t it specify that the distance of 3cm is perpendicular?
There are many points 3cm from the line INSIDE THE STADIUM, no? (If you measure at an angle).
E.g. Draw a 3cm line perpendicular to, and north of, the line. Then from the base of that line, draw another 3cm line at an angle of 45 degrees to the line. The point where this new line ends will be more than 1cm inside the stadium.

Doesn’t that mean that your original shape, your stadium in this case, doesn’t actually cover THE LOCUS OF ALL POINTS 3CM FROM THE LINE?

 

[BvU]

I was asked to draw THE LOCUS OF ALL POINTS 3cm from a line.

That generally means perpendicular distance (a line segment of 3 cm fits between the point and the line)

I don't think a mathematical line is of finite length ... so you might ask yourself what the definition is in the context of your exercise.

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[Mark44]

I agree with @BvU that distance means the perpendicular distance, and that a mathematical line has no endpoints. The intended solution of the problem is two parallel lines on either side of the given line, and 3 cm. away from it.

 

[paulb203]

That generally means perpendicular distance (a line segment of 3 cm fits between the point and the line)

I don't think a mathematical line is of finite length ... so you might ask yourself what the definition is in the context of your exercise.

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Thanks, BvU.
Yeah, I should have said line segment, and not line.
And I was pretty sure that they meant perpendicular distance, and pretty sure what shape they were after; but I'm still wondering if the question is imprecise, especially for the 'uninitiated'. How is one to know that they mean perpendicular distance, without being told? Or is this akin to axioms (there are certain assumptions in maths that don't have to be spelled out)?

 

[paulb203]

I agree with @BvU that distance means the perpendicular distance, and that a mathematical line has no endpoints. The intended solution of the problem is two parallel lines on either side of the given line, and 3 cm. away from it.

Thanks, Mark44

 

[paulb203]

Thanks, BvU.
Yeah, I should have said line segment, and not line.
And I was pretty sure that they meant perpendicular distance, and pretty sure what shape they were after; but I'm still wondering if the question is imprecise, especially for the 'uninitiated'. How is one to know that they mean perpendicular distance, without being told? Or is this akin to axioms (there are certain assumptions in maths that don't have to be spelled out)?

P.S.
I'm less interested in the correct answer from the homework point of view, and more interested in the answer generally (that's why I didn't post it here).

 

[BvU]

P.S.
I'm less interested in the correct answer from the homework point of view, and more interested in the answer generally (that's why I didn't post it here).

Good !

But I have little to add to what I posted in #2.
In the context of geometry, 'distance' is perpendicular distance.

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[Mark44]

I'm still wondering if the question is imprecise, especially for the 'uninitiated'.

IMO, no, the question is not imprecise or ambiguous.

How is one to know that they mean perpendicular distance, without being told?

Because any distance other than the perpendicular distance from a point to a line is going to be longer than the perpendicular distance. The "uninitiated" might after some thought realize that there is a minimum distance between a point and a line.

 

[paulb203]

Thanks, Mark

“Because any distance other than the perpendicular distance from a point to a line is going to be longer than the perpendicular distance.”

I’m not talking about a non-perpendicular distance from the given line segment to the parallel line I drew above it; I’m talking about a non-perpendicular distance of 3cm from the given line segment (which wouldn’t reach the parallel line, and would therefore be inside the stadium.

“The "uninitiated" might after some thought realize that there is a minimum distance between a point and a line.”

I get that, but I’m don’t see how it affects my point.

Would the following be more precise;

-Draw the locus of all points WHOSE SHORTEST DISTANCE FROM THE GIVEN LINE SEGMENT IS 3cm

 

[Mark44]

Draw the locus of all points WHOSE SHORTEST DISTANCE FROM THE GIVEN LINE SEGMENT IS 3cm

Or "whose minimal distance from the given line segment is 3 cm."
Either of these would be clearer than the original phrasing.

 

[Vanadium 50]

How is one to know that they mean perpendicular distance, without being told?

Do you not have a definition of distance between a point and a line? If so, you have been told.

 

[paulb203]

Do you not have a definition of distance between a point and a line? If so, you have been told.

Thanks, Vanadium.

I’m not sure. I’m guessing that if asked what the distance between a point and a line was I would ask, ‘Where on the line’?

I know the shortest distance between two points is a straight line.

I know now, from the guys on this thread, that teachers etc generally mean the perpendicular distance regards this type of question.

But I do think it would help students at GCSE level to state that they mean that; as I was left wondering, ‘What about all the other points 3cm from the line (segment), the points at the end of non-perpendicular 3cm line segments from the main line segment.

 

[Vanadium 50]

Every textbook I have seen defines this. The definitions may be different but equivalent (perpendicular distance or shortest distance) but the whole point of Geometry is to make clear arguments starting from definitions.