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**Proof using distance, midpoint, and slope**

Coordinate Geometry Proof using distance, midpoint, and slope

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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

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**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

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**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

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**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.

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**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.

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