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.
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