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Quadrilaterals

If M and ...

Question

If $M$ and $N$ are mid points of the diagonals $A C$ and $B D$ respectively of a quadrilateral $A B C D$ show that $\to A B + \to A D + \to C B + \to C D = 4 \to M N$ .

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Solution

We have,

$M$ and $N$ are the midpoints of the diagonals $A C$ and $B D$ $ respectively.

Let

$\to A = \to a$

$\to B = \to b$

$\to C = \to c$

$\to D = \to d$

Now,

$- \to M = \frac{\to a + \to c}{2}$

$\to N = \frac{\to b + \to d}{2}$

L.H.S.

$- - \to A B + - - \to A D + - - \to C B + - - \to C D = 4 - -- \to M N$

$\Rightarrow \to b - \to a + \to d - \to a + \to b - \to c + \to d - \to c$

$\Rightarrow 2 \to d - 2 \to a + 2 \to b - 2 \to c$

$\Rightarrow 2 (\to b + \to d) - 2 (\to a + \to c)$

$\Rightarrow 4 N - 4 M$

$\Rightarrow 4 (N - M)$

$\Rightarrow 4 M N$

$R . H . S .$
Hence, proved.

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