Prove that when the points A(7,3) B(9,6) C(10,12) D(8,9) are joined in order, then they will form a parallelogram
Answers
Given : points A(7,3) B(9,6) C(10,12) D(8,9) are joined in order
To Find : Prove that ABCD is a parallelogram
Solution:
We can use any of the following to show if ABCD is a parallelogram :
Two pairs of opposite sides are parallel
Two pairs of opposite sides are equal in length.
The diagonals bisect each other.
Easiest is diagonals bisect each other.
Diagonal AC A(7,3) C(10,12)
Mid point = ( 7 + 10)/2, (3 + 12)/2
= 17/2 , 15/2
Diagonal BD = B(9,6) D(8,9)
Mid point = ( 9 + 8)/2, (6 +9)/2
= 17/2 , 15/2
Mid point of both Diagonal is same Hence
Diagonal bisect each other
hence ABCD is a parallelogram
QED
Hence Proved
Another way
Two pairs of opposite sides are equal in length.
AB = CD = √13
BC = AD = √37
QED
one more way :
Two pairs of opposite sides are parallel
Slope of AB = Slope of CD = 3/2
Slope of BC = slope of AD = 6
QED
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