Transcript 6.3 Proving Quadrilaterals are Parallelograms

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
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
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
C(6, 5)
4
B(1, 3)
2
D(7, 1)
5
A(2, -1)
-2
-4
http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.3%20Proving%20Qu
ads%20are%20Parallelograms.ppt
1
Ex: Using properties of parallelograms


Method 1—Show that opposite
sides have the same slope, so
they are parallel.
Slope of AB.


1–5=-4
7–6
B(1, 3)
2
D(7, 1)
5–3=2
6 -1 5
Slope of DA.


4
Slope of BC.


C(6, 5)
Slope of CD.


3-(-1) = - 4
1-2
6
-1–1=2
2- 7 5
AB and CD have the same
slope, so they are parallel.
Similarly, BC ║ DA.
5
A(2, -1)
-2
-4
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
1
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
6
C(6, 5)
4
B(1, 3)
2
D(7, 1)
5

AB ≅ CD and BC ≅ DA.
Because both pairs of opposites
sides are congruent, ABCD is a
parallelogram.
A(2, -1)
-2
-4
http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.3%20Proving%20Qu
ads%20are%20Parallelograms.ppt
1
Ex: Using properties of parallelograms
 Method 3—Show that
one pair of opposite
sides is congruent and
B(1, 3)
parallel.
6
C(6, 5)
4


Slope of AB = Slope of CD
= -4
AB=CD = √17
2
D(7, 1)
5
A(2, -1)
-2

AB and CD are congruent
and parallel, so ABCD is a
parallelogram.
-4
http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.3%20Proving%20Qu
ads%20are%20Parallelograms.ppt
1
Ex: Using properties of trapezoids


Show that ABCD is a trapezoid.
Compare the slopes of opposite sides.





C(4, 7)
6
B(0, 5)
4
The slopes of AB and CD are equal, so AB ║
CD.


The slope of AB = 5 – 0 = 5 = - 1
0 – 5 -5
The slope of CD = 4 – 7 = -3 = - 1
7–4 3
8
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.
D(7, 4)
2
5
A(5, 0)
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