Geometry of the Line (1) The MidPoint of the Line

Well I'm not sure that I'll bother to start with a definition of a line, although Barry Mazur, in his book that tied up my guts, Imagining Numbers, quotes Euclid: "one and only one line passes through any two distinct points on the plane". That makes sense to me.

For the purposes of this post the line will be assumed to be straight, and two dimensional. We are not going to muck about with curves and stuff.

So.

We have already seen that if you know two co-ordinates on the line you can calculate its gradient. This is very difficult to do with only one set of co-ordinates, unless you have a perpendicular line handy (which we don't, at the moment - that might come later).




If your line begins at 2,2 and ends at 8,4, as above, you need to think "How far has it gone x-wise and y-wise?" Well it has gone 6 units in the x direction and 2 in the y direction.

To find the midpoint of this line you can either just say well mid-way through a movement of 6 is 3, and mid-way through a movement of 2 is 1. So 2+3 = 5 (the x co-ordinate of your mid-point) and 2+1=3 (the y co-ordinate). Your midpoint is (5,3).

This isn't practical for a lot of lines which are (3 1/2, 5 1/2) and so on. So instead we use a simple formula:

The midpoint is, (x₁ + x₂)/2, (y₁+y₂)/2

So: (2+8)/2, (2+4)/2

Thus: 5, 3

The midpoint is (5,3), as above.


The midpoint is useful for all sorts of knotty problems involving perpendicular bisectors and stuff.