Geometric proof help!!!?

Question

Given: A (-3, 6)
B (6, 0)
C (9, -9)
D (0, -3)
Prove ABCD is a parallelogram, but not a rhombus

Please help!!!! Going crazy!!

Answer 1

find the slopes of AB, BC, CD and DA with two point formula

m = (y1-y2)/(x1-x2)

m of AB = (6-0)/(-3-6) = -2/3
m of BC = -3
m of CD = -2/3
m of DA = -3
we know that when slopes of lines are equal they are parallel
so AB is parallel to CD
and BC is parallel to DA
the opposite sides of the Quadrilateral ABCD are parallel
its definitely a parallelogram

now distance between the points
distance = sqrt( (x1-x2)^2 + (y1-y2)^2 )
length of AB = sqrt( (-3-6)^2 + (6-0)^2 ) = sqrt(117)
but BC = sqrt(45)
though opposite side are parallel, the two adjacent sides are not equal in length
hence ABCD is a parallelogram, but not a rhombus

Answer 2

Parallelogram is a quadrilateral that has the opposite sides of equal length. Rhombus is a quadrilateral that has all sides of equal length.

The coordinates give a figure with the sides: AB, BC, CD, DA.
All we have to do is to prove that AB=CD and BC=DA, but not all four sides are equal.

Calculate the length of the four sides. Maybe it is helpful, if you imagine a right-angled triangle above the sides. In this case you have a right-angled triangle, the hypotenuse is the side of the parallelogram, the other two sides are given by the difference of the respective coordinates of the given points.

It means that in case of the side AB, the two shorter sides of the triangle will be Ax-Bx (x coordinate of A minus x coordinate of B) and Ay-By (y coordinate of A minus y coordinate of B).

Pythagoras' Theorem allows you to calculate the hypotenuse (the side of the parallelogram) from the two other sides.

Thus:
AB^2 = (Ax-Bx)^2 + (Ay-By)^2
AB^2 = (-3-6)^2 + (6-0)^2
AB^2 = (-9)^2 + 6^2
AB^2 = 81+36
AB^2 = 117
AB^2 = sqrt(117)

(a^2 means the square of a, sqrt(a) means the square root of a)

Repeating the calculations with the three other sides, you get:
AB=sqrt(117)
BC=sqrt(90)
CD=sqrt(117)
DA=sqrt(90)

Thus AB=CD, BD=DA, but AB is not equal to BC or DA, so the figure is a parallelogram.

Answer 3

Don't go crazy, here is what you can do.

Look at points A & B.
The difference between the y-components is -6.
This is the "run" of segment AB.
The difference between the x-components is 9.
This is the "rise" of segment AB.

Now, look at the opposite side of the quadrilateral, segment DC.
The difference between the y-components is -6.
This is the "run" of segment DC.
The difference between the x-components is 9.
This is the "rise" of segment DC.

Okay now, segment AB has the same rise as DC, and the same run as DC.
Therefore these two segments will be parallel, and will also be equal in length.
So, regardless of their locations in the x-y plane, The two missing sides BC, and AD, will also be parallel to each other, and equal in length.

Now, we only need to show that AB and BC are different in length.
Notice that the run of BC is 9.
Recall that the rise of AB is also 9
So, in order for BC to be the same length as AB, the rise of BC will have to equal the run of AB (which we already know is -6).
The rise of BC is -3, so BC cannot be equal to AB in length.

ABCD is a parallelogram, but not a rhombus.

Answer 4

In a nutshell, we are trying to establish that the two sides of the diagram are of the same gradient and the other two are of the same gradient too (proving parallelogram) and to also establish that one side is longer/shorter than the other (proving it is not a rhombus).

from this point gradient of AB would be written as mAB.

[gradient = (y' - y)/(x' - x)]

mAB = (6-0)/(-3-6)
mAB = -2/3

mCD = (-3-(-9))/(0-9)
mCD = -2/3

therefore, line AB and CD are parallel.

mBC = (0- -9)/(6-9)
mBC = -3

mDA = (6- (-3))/(-3 - 0)
mDA = -3

line BC and DA are parallel. Therefore, ABCD is a parallelogram.
============================================

[distance = sqrt [(x'-x)² + (y'-y)²] ]

length of line AB = sqrt [(-3-6)² + (6-0)²]
length of line AB = sqrt (117)

length of line BC = sqrt [(6-9)² + (0- -9)²]
length of line BC = sqrt (90)

since length AB does not equal length of line BC, the diagram is not a rhombus.

Answer 5

AB has a slope of -6/9 = -2/3
CD has a slope of -6/9 = -2/3Thus AB || CD
BC has a slope of -9/3 = - 3
AD has a slope of 9/-3 = -3
Thus AD || BC
Thus ABCD is a ||=ogram
AB = sqrt(6^2+3^2) = sqrt(45)
BC = sqrt(9^2+3^2) = sqrt(90)
Thus ABCD is a ||-ogram and not a rhombus because its sides are not all equal