1. Using Coordinate Geometry to Prove Parallelograms

2. 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

3. 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.

4. 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

5. 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.

6. 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.

7. 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 

8. 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.

9. 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.

10. 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.