it is geometry
Hi, Olivia:
1) Consider two points on the number line. As an example, let's take points x=0 and x=2. It is obvious, that the midpoint coordinate is x=1;
2) What if we have x=4 and x=9, for example? Again, the midpoint is x=6.5, since its distance from both points is the same. Indeed, |4-6.5|=|-2.5|=2.5 and |9-6.5|=|2.5|=2.5. Remember, distance is the absolute value of the difference between x-coordinates of the first and the second point.
3) Can we find a formula to obtain the x-coordinate of the midpoint? Let us try.
Let the first point be x_{1} and the second be x_{2}. Let the midpoint be x_{m}. Without losing any generality, let us assume x_{1}<x_{2}. Then x_{1}<x_{m}<x_{2}, since midpoint shall be between two points. We know that distance from midpoint to each of two points is the same. This means that:
x_{m}-x_{1}=x_{2}-x_{m}; Solve for x_{m},
2x_{m}=x_{2}+x_{1};
x_{m}=(x_{1}+x_{2})/2;
So the coordinate of a midpoint is just the average of two coordinates.
Now, we can generalize. If we consider two points on the plane, with coordinates (x_{1},y_{1}) and (x_{2},y_{2}), it is natural to suggest that the coordinates of a midpoint will be given by the average of respective y- and x-coordinates of endpoints. So, we suggest that:
(x_{m},y_{m})=(½(x_{1}+x_{2});½(y_{1}+y_{2}));
And this is indeed the case. So in your example,
x_{m}=½(6+4)=5;
y_{m}=½(7+3)=5;
or
(x_{m},y_{m})=(5,5)
As a matter of fact, this can be generalized for 3-dimensional case and for any number of dimensions.
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