Quadrilaterals in the Coordinate Plane

I can find the slope and distance between two points

I can use the properties of quadrilaterals to prove that a figure in the coordinate plane is a parallelogram, rhombus, rectangle, square, or trapezoid.

ExampleGiven points:

A(0,2) B(3,6) C(8,6) D(5,2)

How can we prove ABCD is a parallelogram, rectangle, rhombus, or a square?

Slope FormulaGiven: (x1,y1) (x2,y2)

*Perpendicular lines have opposite reciprocal slopesex: m = 3/2 & -2/3

*Parallel lines have the same slopeex: m = 3 & 3

2 1

2 1

y y y risem

x x x run

ExampleA(0,2) B(3,6) C(8,6) D(5,2)

Step 1: Is ABCD a parallelogram?

*check to see if slopes are parallel

2 1

2 1

y ym

x x

AB

DC

BC

AD

ExampleA(0,2) B(3,6) C(8,6) D(5,2)

Step 2: Is ABCD a rectangle or a square?

*were the slopes perpendicular?

Distance FormulaGiven: (x1,y1) (x2,y2)

OR….

Use the Pythagorean Theorem on a grid

2 2

2 1 2 1d y y x x

2 2 2a b c

ExampleA(0,2) B(3,6) C(8,6) D(5,2)

Step 3: Is ABCD a rhombus?

*check the lengths

AB

BC

2 2

2 1 2 1d y y x x