1. Coordinate Geometry Adapted from the Geometry Presentation by Mrs. Spitz Spring 2005 http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.3%20Proving%20Qu ads%20are%20Parallelograms.ppt

2. Using Coordinate Geometry When a figure is in the coordinate plane, you can use the Distance Formula to prove that sides are congruent and you can use the Slope Formula to prove sides are parallel or perpendicular. http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.3%20Proving%20Qu ads%20are%20Parallelograms.ppt

3. Ex: Using properties of parallelograms Show that A(2, -1), B(1, 3), C(6, 5) and D(7,1) are the vertices of a parallelogram. http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.3%20Proving%20Qu ads%20are%20Parallelograms.ppt

4. Ex: Using properties of parallelograms Method 1—Show that opposite sides have the same slope, so they are parallel. Slope of AB.  3-(-1) = - 4 1 - 2 Slope of CD.  1 – 5 = - 4 7 – 6 Slope of BC.  5 – 3 = 2 6 - 1 5 Slope of DA.  - 1 – 1 = 2 2 - 7 5 AB and CD have the same slope, so they are parallel. Similarly, BC ║ DA. Because opposite sides are parallel, ABCD is a parallelogram. http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.3%20Proving%20Qu ads%20are%20Parallelograms.ppt

5. Ex: Using properties of parallelograms Method 2—Show that opposite sides have the same length. AB=√(1 – 2) 2 + [3 – (- 1) 2 ] = √17 CD=√(7 – 6) 2 + (1 - 5) 2 = √17 BC=√(6 – 1) 2 + (5 - 3) 2 = √29 DA= √(2 – 7) 2 + (-1 - 1) 2 = √29 AB ≅ CD and BC ≅ DA. Because both pairs of opposites sides are congruent, ABCD is a parallelogram. http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.3%20Proving%20Qu ads%20are%20Parallelograms.ppt

6. Ex: Using properties of parallelograms Method 3—Show that one pair of opposite sides is congruent and parallel. Slope of AB = Slope of CD = -4 AB=CD = √17 AB and CD are congruent and parallel, so ABCD is a parallelogram. http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.3%20Proving%20Qu ads%20are%20Parallelograms.ppt

7. Ex: Using properties of trapezoids Show that ABCD is a trapezoid. Compare the slopes of opposite sides.  The slope of AB = 5 – 0 = 5 = - 1 0 – 5 -5  The slope of CD = 4 – 7 = -3 = - 1 7 – 4 3 The slopes of AB and CD are equal, so AB ║ CD.  The slope of BC = 7 – 5 = 2 = 1 4 – 0 4 2  The slope of AD = 4 – 0 = 4 = 2 7 – 5 2 The slopes of BC and AD are not equal, so BC is not parallel to AD. So, because AB ║ CD and BC is not parallel to AD, ABCD is a trapezoid. http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.5%20Trapezoids.ppt

8. Homework Work Packet: Coordinate Geometry #1, 3, 4 Find all 4 slopes, all 4 distances, and name the figure