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Addition of Vectors

In quadrilate...

Question

In quadrilateral ABCD, prove that
(i) $¯ ¯¯¯¯¯¯ ¯ A B + ¯ ¯¯¯¯¯¯ ¯ B C + ¯ ¯¯¯¯¯¯¯ ¯ A D + ¯ ¯¯¯¯¯¯¯ ¯ D C = 2 ¯ ¯¯¯¯¯¯ ¯ A C$
(ii) $¯ ¯¯¯¯¯¯¯ ¯ A D + ¯ ¯¯¯¯¯¯ ¯ B C = 2 ¯ ¯¯¯¯¯¯¯¯¯ ¯ M N$ , where M and N are the mid points of side AB and CD respectively.

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Solution

$(i)$ $\to A B + \to B C = \to A C \to (1)$
$\to A D + \to D C = \to A C \to (2)$
Adding (1) and (2) ,
$\to A B + \to B C + \to A D + \to D C = 2 \to A C$
$(i i)$ $\to M N + \to N C = \to M C, \to M B + \to B C = \to M C$
$\to M N + \to N C = \to M B + \to B C \to (1)$
$\to M A + \to A D = \to M D, \to M N + \to N D = \to M D$
$\to M A + \to A D = \to M N + \to N D \to (2)$
From (1) , $\to M N - \to N D = \to M B + \to B C [∵ \to N C = - \to N D]$
$\to M N - \to M B = \to N D + \to B C$
$\to M N + \to M A = \to N D + \to B C [∵ \to M A = - \to M B]$
$\to M N + \to M N + \to N D - \to A D = \to N D + \to B C$ [From (2), $\to M A = \to M N + \to N D$ ]
$2 \to M N + \to N D = \to N D + \to A D + \to B C$
$2 \to M N = \to A D + \to B C$

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