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We begin with two points.
These are the two points that we are interested in finding the midpoint for.
And checking, of course.
To make this easier, we connect these two points with a line segment.
Every line segment is somewhere in space,
so we can draw a Cartesian plane to represent where our line segment is.
We draw the two other sides, besides the hypotenuse, of a right triangle,
having the right angle connect the x- and y-axes.
This can be used to find the midpoint, based on what the halfway points of the x- and y-axes are.
We label the first point on our line segment as Point A,
and the second point as Point B.
We label the origin of our Cartesian plane as C.
We label the midpoint between A and C as X, because it is on the x-axis,
and the midpoint between B and C as Y because it is on the y-axis.
We now draw a square with vertices X, C, and Y;
with the final vertex being labelled M.
This is the midpoint of AB.
An interesting thing you should notice now is that our large triangle,
consisting of the x-axis, y-axis, and our line segment,
has been broken up into two triangle and a square.
Triangles AMX and BMY are congruent,
due to the fact that the central portion of the triangle we've drawn,
between the x-axis, y-axis, and our line segment is a square.
This means that line segments AM and BM are the same length,
which means that M is the midpoint of A and B,
because it lies in the middle.
Point A's location on the x-axis can be labelled (x1),
and it's location on the y-axis can be labelled (y1).
Point B's location on the x-axis can be labelled (x2),
and it's location on the y-axis can be labelled (y2).
Point X is in the middle of Points x1 and x2,
which means it is at the average of these or X=(x2-x1)/2.
Point Y lies at the average of Points y1 and y2,
which means Y=(y2-y1)/2.
Point M has an x value of X and a y value of Y,
which means M is (x2-x1)/2,(y2-y1)/2; the midpoint formula.