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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