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Section Formula

ABCD is a qua...

Question

$A B C D$ is a quadrilateral, $E$ is the point of intersection of the line joining the middle points of the opposite sides. lf $O$ is any point, then $^O A +^O B +^O C +^O D =$

4^O E

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3^O E

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2^O E

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^O E

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Solution

The correct option is A $4^O E$
Let $P V$ (position vector) of $A, B, C, D$ be $\to a, \to b, \to c, \to d$ respectively.
$P V$ of $P$ $=$ $\frac{\to a + \to b}{2}$
$P V$ of $Q$ $=$ $\frac{\to c + \to d}{2}$
$P V$ of $v$ $=$ $\frac{\to a + \to d}{2}$
$P V$ of $u$ $=$ $\frac{\to b + \to b}{2}$
$\to P Q = \frac{1}{2} (\to d - \to a + \to c - \to b)$
$\to u v = \frac{1}{2} (\to a - \to b + \to d - \to c)$
So, $E = \frac{\to a + \to b + \to + c + \to d}{4}$
Let $O$ be origin, so $\to O A + \to O B + \to O C + \to O D = \to a + \to b + \to c + \to d$
Therefore, $4 \to O E = \to a + \to b + \to c + \to d$

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