Math Is Fun Forum

paulb203

Member

Registered: 2023-02-24

Posts: 321

Loci and Euclid

I was asked to draw the LOCUS OF ALL POINTS which are equidistant from points A AND B.

A and B were two points 10 or so cm apart halfway down a page of A4.

I drew a perpendicular bisector connecting my compass markings to get the halfway mark between the points, which was a line of 10cm or so.

But then, as I’m new to this, I wondered, if the line is to be the locus of ALL points equidistant from A and B, should extend to the top and bottom of the page?

Then I wondered if it should extend infinitely in both directions, like one of those Euclidean theoretical (idealised?) lines of infinite length.

Am I on the right track in my thinking here?

P.S. The answer (this was a Corbettmaths question) showed a line extending beyond the compass markings, but not to the top or bottom of the page.

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Prioritise. Persevere. No pain, no gain.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Loci and Euclid

Using compasses to construct the perpendicular bisector is the correct method. (You could also measure for the midpoint and use a protractor for the perpendicular line but see note below about awarding marks for the construction)

In theory the line extends to infinity in both directions.  Clearly, in practice, you have to end the line somewhere. A marker would be looking for your construction so don't erase anything.  To show you understand the locus doesn't end where your arcs intersected extend the line beyond those points.

If you really want to be mathematically pedantic you could add a note that your locus extends to infinity.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

paulb203

Member

Registered: 2023-02-24

Posts: 321

Re: Loci and Euclid

Thanks, Bob.

Fascinating stuff (lines of infinite lengths, etc).

The very next question involved a ray!

It asked for the LOCUS OF ALL POINTS which are equidistant from lines CD and CE, which formed and angle of around 100 degrees.

I constructed my angular bisector and started drawing the ray, again thinking, this would extend infinitely if Euclid had anything to do with it.

So I drew it to the end of the page, but imagined it continuing across my desk, the rest of the room, through the wall, and, 'To infinity and beyond!'

P.S. I've done my GCSE Maths now (just waiting for the result), so I'm doing this for fun now (and with a view to maybe trying A Level).

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Prioritise. Persevere. No pain, no gain.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Loci and Euclid

Did you consider the direction of your infinite line?  I have received a hyperwave transmission from Alpha Centauri complaining about an infinite line that keeps intersecting their planet (every 24 hours) and, crucially, is interfering with their reception of House* on the interstellar carrier transmitter/receiver.  They believe the line originated in your area.

* They are big fans of Hugh Laurie.

It may only be imaginary to you, but, in interstellar carrier transmission/reception, imaginary lines play a significant role.  Indeed, without complex numbers it's doubtful such things would be possible

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

paulb203

Member

Registered: 2023-02-24

Posts: 321

Re: Loci and Euclid

Lol!

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Prioritise. Persevere. No pain, no gain.