COORDINATE GEOMETRY PROOFS USE OF FORMULAS TO PROVE

COORDINATE GEOMETRY PROOFS USE OF FORMULAS TO PROVE STATEMENTS ARE TRUE/NOT TRUE: Distance: d= Midpoint: midpoint= ( Slope: m = )

Distance Formula l. Distance: d= When two line segments have the same distance, they are equal in length.

Midpoint Formula l. Midpoint = When two line segments have the same midpoint, it shows that they are bisecting each other.

Slope Formula l. Slope: m= When two lines have equal slopes, they are parallel. When two lines have slopes which are negative reciprocals, they are perpendicular.

Proving a Quadrilateral is a Parallelogram Methods: 1. Show that both pairs of opposite sides are equal. l 2. l 3. l 4. l Use distance formula for 4 sides Show that both pairs of opposite sides are parallel. Use slope formula for 4 sides Show that diagonals bisect each other. Use midpoint formula for 2 diagonals 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)