Q. 74.4( 7 Votes )

# If (x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3}) and (x_{4}, y_{4}) point s are joined in order to form a parallelogram, then prove that x_{1} + x_{3} = x_{2} + x_{4} and y_{1} + y_{3} = y_{2} + y_{4}.

Answer :

A(x_{1}, y_{1}), B(x_{2}, y_{2}), C(x_{3}, y_{3}) and D(x_{4}, y_{4})

We know that a quadrilateral is a parallelogram if the co-ordinates of mid-point s of its both the diagonals are same.

Therefore, we’ll find the mid-point s of diagonal AC and BD.

Let the co-ordinates of mid-point of AC be (x_{0}, y_{0}).

And, we know mid-point formula, i.e. the coordinates of mid-point of line joining (x_{1}, y_{1}) and (x_{2}, y_{2}) is

Similarly, let the co-ordinates of mid-point of AC be (x_{5}, y_{5}).

And, since it is a mid-point –

Now, since ABCD is a parallelogram –

⇒ (x_{0}, y_{0}) = (x_{5}, y_{5})

⇒ x_{0} = x_{5} and y_{0} = y_{5}

⇒x_{1} + x_{3} = x_{2} + x_{4} and y_{1} + y_{3} = y_{2} + y_{4}

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