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**Using Coordinate Geometry to Prove Parallelograms**

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**Using Coordinate Geometry to Prove Parallelograms**

Definition of Parallelogram Both Pairs of Opposite Sides Congruent One Pair of Opposite Sides Both Parallel and Congruent Diagonals Bisect Each Other

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**Definition of a Parallelogram**

Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3,1) . I need to show that both pairs of opposite sides are parallel by showing that their slopes are equal.

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**Definition of a Parallelogram**

Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3,1) . AB: m = 6 – 0 = 6 = – CD: m = 1 – 7 = = – BC: m = 7 – 6 = – AD: m = 1 – 0 = – AB ll CD BC ll AD ABCD is a Parallelogram by Definition

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**Both Pairs of Opposite Sides Congruent**

Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3,1) . I need to show that both pairs of opposite sides are congruent by using the distance formula to find their lengths.

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**Both Pairs of Opposite Sides Congruent**

Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3,1) . AB = (2 – 0)2 + (6 – 0)2 = = 40 CD = (3 – 5)2 + (1 – 7)2 = = 40 AB CD BC = (5 – 2)2 + (7 – 6)2 = = 10 AD = (3 – 0)2 + (1 – 0)2 = = 10 BC AD ABCD is a Parallelogram because both pair of opposite sides are congruent.

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**One Pair of Opposite Sides Both Parallel and Congruent**

Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3,1) . I need to show that one pair of opposite sides are both parallel and congruent. ll and

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**One Pair of Opposite Sides Both Parallel and Congruent**

Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3,1) . BC: m = 7 – 6 = – AD: m = 1 – 0 = – BC ll AD BC = (5 – 2)2 + (7 – 6)2 = = 10 AD = (3 – 0)2 + (1 – 0)2 = = 10 BC AD ABCD is a Parallelogram because one pair of opposite sides are parallel and congruent.

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**Diagonals Bisect Each Other**

Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3,1) . I need to show that each diagonal shares the SAME midpoint.

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**Diagonals Bisect Each Other**

Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3,1) . 5 , The midpoint of AC is , 5 , The midpoint of BD is , ABCD is a Parallelogram because the diagonals share the same midpoint, thus bisecting each other.

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