If (x1, y1), (x2, y2), (x3, y3) and (x4, y4) point s are joined in order to form a parallelogram, then prove that x1 + x3 = x2 + x4 and y1 + y3 = y2 + y4.
A(x1, y1), B(x2, y2), C(x3, y3) and D(x4, y4)
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 (x0, y0).
And, we know mid-point formula, i.e. the coordinates of mid-point of line joining (x1, y1) and (x2, y2) is
Similarly, let the co-ordinates of mid-point of AC be (x5, y5).
And, since it is a mid-point –
Now, since ABCD is a parallelogram –
⇒ (x0, y0) = (x5, y5)
⇒ x0 = x5 and y0 = y5
⇒x1 + x3 = x2 + x4 and y1 + y3 = y2 + y4
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