1. Proof using distance, midpoint, and slope
Coordinate Geometry
Proof using distance, midpoint, and slope

2. Coordinate proof
Prove a quadrilateral is a parallelogram using three different methods.
Method One: The slope formula
Method Two: The distance formula
Method Three: The midpoint formula

3. Proving a quadrilateral is a Parallelogram
Given quadrilateral ABCD, with vertices A(1,2), B(6,5),
C(7,2), and D(2,-1). Prove ABCD is a parallelogram x y A B C D

4. Method 1: the slope formula
Given quadrilateral ABCD, with vertices A(1,2), B(6,5),
C(7,2), and D(2,-1), prove ABCD is a parallelogram
Conclusion:
Since opposite sides of quadrilateral ABCD have the same slope,
they are parallel. Quadrilateral ABCD has two pairs of parallel sides,
therefore it is a parallelogram

5. Method 2: the distance formula
Given quadrilateral ABCD, with vertices A(1,2), B(6,5),
C(7,2), and D(2,-1), prove ABCD is a parallelogram
Conclusion:
Since opposite sides of quadrilateral ABCD have the same length,
they are congruent. Quadrilateral ABCD has two pairs of opposite,
parallel sides, therefore it is a parallelogram.

6. Method 3: the midpoint formula
Given quadrilateral ABCD, with vertices A(1,2), B(6,5),
C(7,2), and D(2,-1), prove ABCD is a parallelogram
Midpoint of AC
Midpoint of BD
Conclusion:
Since the midpoints of AC and BD are the same point, they must
bisect each other. Quadrilateral ABCD has bisecting diagonals,
therefore it is a parallelogram.