Using Coordinate Geometry to Prove Parallelograms Using Coordinate

Using Coordinate Geometry to Prove Parallelograms

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

L 34 Opener QUAD JOHN J(-3, 1) O(3, 3) H(5, 7) N(-1, 5) a) Use the slope formula and calculate the slope of all four sides. b) Write down and fill in the blanks Segments with the same slopes are _____. Since opposite sides are _____ then QUAD JOHN is a _______ by the ________. c) Graph QUAD JOHN and check that it is a parallelogram.

Definition of a Parallelogram Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B 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. A C D

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

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). C B I need to show that both pairs of opposite sides are congruent by using the distance formula to find their lengths. A D

Both Pairs of Opposite Sides Congruent A 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 CD = (3 – 5)2 + (1 – 7)2 = 4 + 36 = 40 AB CD BC = (5 – 2)2 + (7 – 6)2 = 9 + 1 = 10 C B D = 4 + 36 = 40 AD = (3 – 0)2 + (1 – 0)2 = 9 + 1 = 10 BC AD ABCD is a Parallelogram because both pair of opposite sides are congruent.

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). C B I need to show that one pair of opposite sides is both parallel and congruent. ll (slope) and (distance) A D

One Pair of Opposite Sides Both B 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 BC ll AD BC = (5 – 2)2 + (7 – 6)2 = 9 + 1 = 10 A C D AD: m = 1 – 0 = 1 3– 0 3 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.

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). B I need to show that each diagonal shares the SAME midpoint. A C D

Diagonals Bisect Each Other C B 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). A The midpoint of AC is 0+5 , 0+7 2 2 5 , 7 2 2 The midpoint of BD is 2+3 , 6+1 2 2 5 , 7 2 2 ABCD is a Parallelogram because the diagonals share the same midpoint, thus bisecting each other. D