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You need to remember the formula of a midpoint. Notice that this formula averages the coordinates of the points that stand at the left and the right of the midpoint such that:
`x = (x_A + x_B)/2`
For example, we have the points A(2,5) and the point B(4,3) and we need to find the point in between.
We will average both x and y coordinates to find the midpoint.
Let the midpoint be M(xM, yM).
==> xM = (xA+xB)/2 = (2+4)/2 = 6/2 = 3
==> yM = (yA+yB)/2 = (5+3)/2 = 8/2 = 4
Then, the midpoint is given by:
To average 2 points, we'll have to calculate the arithmetical mean of the correspondent coordinates of the points.
If the point is located on a segment, at the same distance form both endpoints, this point is called endpoint.
Let's take 2 points M(x1,y1) and N(x2,y2).
The midpoint is P(x,y)
The distance PM = PN
PM^2 = (x1 - x)^2 + (y1 - y)^2
PN^2 = (x2 - x)^2 + (y2 - y)^2
(x1 - x)^2 + (y1 - y)^2 = (x2 - x)^2 + (y2 - y)^2
We'll expand the squares and we'll eliminate like terms:
x1^2 - x2^2 + y1^2 - y2^2 = 2xx1 - 2xx2 + 2yy1 - 2yy2
(x1 - x2)(x1 + x2) + (y1 - y2)(y1 + y2) = 2x(x1 - x2) + 2y(y1 - y2)
Comparing, we'll get:
2x(x1 - x2) = (x1 - x2)(x1 + x2)
We'll simplify and we'll get:
2x = (x1 + x2)
x = (x1 + x2)/2
y = (y1 + y2)/2
That's the proof of why you has to average the points!
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