The Coordinate Geometry Proof
**Nicole Mosteller**
**EMAT 6690**

This
proof involves the application
of the Midpoint Formula and the Distance Formula both in Relation
to Coordinate Geometry.

Given a right triangle XYZ with the
median of the hypotenuse labeled M.
__Figure 1:__ Given.

Let's impose our right triangle onto
a coordinate graph.
For simplicity sake, let's
use Quadrant I and place our right angle at (0, 0) so that
the legs of our right triangle fall
along the x-axis and y-axis.
__Figure 2:__ The right triangle with variable coordinates.
****Why is the coordinate of X (0, 2b) and the coordinate
of Y (2a, 0)?** Using the coefficient values of 2 on for these
coordinates allows for algebraic ease later in the proof. Because
a and b are not restricted to the integers, this notation does
not imply respective coordinates are even.

Since we know that point M is the midpoint
of the hypotenuse,
the coordinate of M if found using
the midpoint formula.

__Figure 3:__ The coordinate of M is (a, b).

Now that we have the coordinates of
each point,
let's use the distance formula to find
the distance between the endpoints of
segments XM, YM, and ZM.

__Figure 4:__ Using the known coordinates, use the distance formula

By substitution, we are able to arrive
at the conclusion
XM = YM = ZM.
By definition, point M is equidistant
from points Z, X, and Y.
So, the midpoint M is eqidistant from
all of the vertices of the right triangle.

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