COORDINATE GEOMETRY PROOFS

USE OF FORMULAS TO PROVE STATEMENTS ARE TRUE/NOT

TRUE:

Distance: d=Midpoint: midpoint= ( )Slope: m =

( ) ( )x x y y2 1

2

2 1

2

x x y y2 1 2 1

2 2

,

y y

x x2 1

2 1

Distance Formula

Distance: d=

When two line segments have the same distance, they are equal

in length.

( ) ( )x x y y2 1

2

2 1

2

Midpoint Formula

Midpoint =

When two line segments have the same midpoint,

it shows that they are bisecting each other.

( , )x x y y2 1 2 1

2 2

Slope Formula

Slope: m =

When two lines have equal slopes, they are parallel.

When two lines have slopes which are negative reciprocals, they are perpendicular.

y y

x x2 1

2 1

Proving a Quadrilateral is a ParallelogramMethods:

1. Show that both pairs of opposite sides are equal.

Use distance formula for 4 sides2. Show that both pairs of opposite sides are

parallel. Use slope formula for 4 sides3. Show that diagonals bisect each other. Use midpoint formula for 2 diagonals4. Show one pair of opposite sides equal and

parallel Use distance and slope for 2 opposite sides

Write a conclusion statement:

1. The quadrilateral is a parallelogram since both pairs of opposite sides of the quadrilateral are equal.

2. The quadrilateral is a parallelogram since both pairs of opposite sides of the quadrilateral are parallel because their slopes are equal.

3. The quadrilateral is a parallelogram since the diagonals bisect each other because they have the same midpoint.

4. The quadrilateral is a parallelogram since it has one pair of opposite sides which are equal and parallel because they have the same slope.

Examples

1. Prove that quadrilateral ABCD is a parallelogram if the coordinates of A( 2 , 3 ), B(8,4 ), C(7, -6 ), and D(1, -7 )

2. Prove quadrilateral JKLM is a parallelogram if the coordinates are: J (0,0), K(a,0), L(a+b,c), and M(b,c).

Homework:

Prove each of the following quadrilaterals are parallelograms. Be sure to use each method which corresponds to the question number.

1. A ( -3,-2 ) B ( 2,-2) C ( 4,1 ) D ( -1,1 )

2. P ( 4,9 ) Q( 6,12) R ( 5,8) S ( 3,5)

3. J ( 1,-3) K ( 1,4) L ( 6,8 ) M ( 6,1)

4. M ( -7,5 ) A ( 2,5 ) T ( 6,-4 ) H ( -3,-4)