Question

if (x1,y1),(X2,y2),(X3,y3)&(x4,y4) points are joined in order to form a parallelogram, then prove that x1+X3=X2+x4 and y1+ y3=y2+y4.​

Answer 1

$\textbf{Given:}$

$\textsf{The points}\;\mathsf{(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)}$

$\textsf{form a parallelogram}$

$\textbf{To prove:}$

$\mathsf{x_1+x_3=x_2+x_4}$

$\mathsf{y_1+y_3=y_2+y_4}$

$\textbf{Solution:}$

$\textsf{Let the given points be}\;\mathsf{A(x_1,y_1),B(x_2,y_2),C(x_3,y_3),D(x_4,y_4)}$

$\textsf{We know that,}$

$\boxed{\textsf{Diagonals of parallelogram bisect each other}}$

$\implies\textsf{Midpoint of diagonal AC=Midpoint of diagonal BD}$

$\implies\mathsf{\left(\dfrac{x_1+x_3}{2},\dfrac{y_1+y_3}{2}\right)=\left(\dfrac{x_2+x_4}{2},\dfrac{y_2+y_4}{2}\right)}$

$\textsf{Equating corresponding co-ordinates, we get}$

$\mathsf{\dfrac{x_1+x_3}{2}=\dfrac{x_2+x_4}{2}}$

$\implies\boxed{\mathsf{x_1+x_3=x_2+x_4}}$

$\mathsf{and}$

$\mathsf{\dfrac{y_1+y_3}{2}=\dfrac{y_2+y_4}{2}}$

$\implies\boxed{\mathsf{y_1+y_3=y_2+y_4}}$

$\textbf{Find more:}$

Show that thepoints A (0, 0), B(3, 0),C(4, 1) and D(1, 1) form a parallelogram​

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