Problem: Prove that quadrilateral METS is a parallelogram given the vertices M ( 0, 0 ), E ( 8, 2 ), T ( 8, 10 ) and S ( 0, 2 )

Possible Solutions: In order to prove a quadrilateral to be a parallelogram, we have three (3) different ways that we can prove this shape is a parallelogram.

1. The first way to do this is by using the midpoint formula two (2) times. If we can find that the diagonals of this shape bisect each other (intersect at the same midpoint), then we will be able to prove this quadrilateral is a parallelogram. In this example, you will perform the midpoint formula on diagonals MT and ES. If you have problems figuring out what the names of the sides and/or diagonals are, make sure you graph the shape first on graph paper. The diagonals are the two point opposite each other in the shape.
2. The second way to do this is by using the slope formula four (4) times. If we can find that opposite sides of this parallelogram have the same slope, then we can prove that opposite sides of this quadrilateral are parallel (have equal slopes). In this example, you will perform the slope formula on sides ME, ET, TS, and SM. If you have problems identifying the four (4) sides of this shape, make sure you graph the figure first, on graph paper.
3. The third way to do this is by using the distance formula four (4) times. If we can find out that opposite sides of this quadrilateral are congruent (same shape, same size), then we can prove that this shape is a parallelogram. In this example, you will perform the distance formula on sides ME, ET, TS, and SM. If you have problems identifying the four (4) sides of this shape, make sure you graph the figure first, on graph paper.

Finally: After you have done all of the math involved in a coordinate geometry proof, you must write a statement that shows your understanding. Writing this statement is the true test for teachers to find out that the student does or does not understand why they have done the math, otherwise we, as teachers, think you may just be blindly following a shell provided by your teacher. This is where we separate the men from the boys. Here are the possible statements for the three different proofs that you may have done here:

1. Quadrilateral METS is a parallelogram. Since the diagonals share a common midpoint, put value here, the diagonals must bisect each other, proving this shape to be a parallelogram.
2. Quadrilateral METS is a parallelogram. Since opposite sides of this quadrilateral have equal slopes, put value here, the opposite sides must be parallel to each other, proving this shape to be a parallelogram.
3. Quadrilateral METS is a parallelogram. Since opposite sides of this quadrilateral have equal distances, put value here, the opposite side must be congruent, proving this shape is a parallelogram.

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