CoordinateGeometry

Adapted from the Geometry Presentation by Mrs. Spitz

Spring 2005http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.3%20Proving%20Quads%20are%20Parallelograms.ppt

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%20Quads%20are%20Parallelograms.ppt

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.

6

4

2

-2

-4

5

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.3%20Proving%20Quads%20are%20Parallelograms.ppt

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.

6

4

2

-2

-4

5

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

Because opposite sides are parallel, ABCD is a parallelogram.

http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.3%20Proving%20Quads%20are%20Parallelograms.ppt

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.

6

4

2

-2

-4

5

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.3%20Proving%20Quads%20are%20Parallelograms.ppt

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.

6

4

2

-2

-4

5

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.3%20Proving%20Quads%20are%20Parallelograms.ppt

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.

8

6

4

2

5 A(5, 0)

D(7, 4)

C(4, 7)

B(0, 5)

http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.5%20Trapezoids.ppt

Homework

Work Packet:

Coordinate Geometry #1, 3, 4

Find all 4 slopes, all 4 distances, and name the figure