Using Coordinate Geometry to Prove Parallelograms

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 3 2 0
2 CD m 1 7 - 6 3 3
5 - 2 BC m 7 6 1 5
2 3 AD m 1 0 1 3
0 3
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 ? 4 36 ?40
CD ?(3 5)2 (1 7)2 ? 4 36 ?40
AB ? CD
BC ?(5 2)2 (7 6)2 ? 9 1 ?10
AD ?(3 0)2 (1 0)2 ? 9 1 ?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 is
both parallel and congruent.
ll (slope) and ? (distance)
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 1 5 2 3
AD m 1 0 1 3 0 3
BC ll AD
BC ?(5 2)2 (7 6)2 ? 9 1 ?10
AD ?(3 0)2 (1 0)2 ? 9 1 ?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 , 72 2
The midpoint of AC is 0 5 , 0 7
2 2
5 , 72 2
The midpoint of BD is 2 3 , 6 1
2 2
ABCD is a Parallelogram because the diagonals
share the same midpoint, thus bisecting each
other.