Points In Between 

Problem1

Point A is (0 , 10), and point B is (6 , 2)

Find the co-ordinates of the point M, which is the mid-point of the line AB.

Point P is on AB, but twice as close to A as to B. Find the co-ordinates of P.

A New Trick

A ‘weighted’ average of two values, or points, works just like a normal (mean) average, except that it takes more of one value than the other…

So, a ‘normal’ average of A and B would be:

    - which gives the mid-point of AB.

Whereas a ‘weighted’ average of the points A and B, giving twice as much weight to A as to B, would be:

   

These should agree with your answers!

Problem 2

The points A and B, taken with the origin C (0, 0), make a triangle.

Something rather surprising happens in any triangle, and you’re going to use these ‘weighted’ averages to see what…

You’ve already found the mid-point of AB – namely M (3, 6).

Now find the mid-points of the other two sides -  N on CB, and P on CA.

Now find the point G1, which is on AN, but twice as close to N as to A.

Repeat this approach, to find the point G2, on BP, but twice as close to P as to B.

And now find G3, on CM, but twice as close to M as to C.

What did you discover?

These three lines, joining each vertex of a triangle to its opposite mid-point, always cross at a single point inside the triangle – two-thirds of the way from each vertex.

The three lines are called the Medians, and the special point is called the ‘Centroid’. 

‘Mazin’

It works for any triangle you can think of. Try one of your own.

Open the File as a Word Document

 

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Last modified: June 18, 2007