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The Mid-Point Theorem

If D, E, F ar...

Question

If $D, E, F$ are respectively the mid points of $- - \to A B, - - \to A C$ and $- - \to B C$ in $Δ A B C,$ then $- - \to B E + - - \to A F =$

A

- - \to D C

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B

\frac{1}{2} - - \to B F

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C

2 - - \to B F

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D

\frac{3 - - \to B F}{2}

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Solution

The correct option is A $- - \to D C$
Let us represent $A, B$ and $C$ with vectors $\to a, \to b$ and $\to c$ respectively.
Here vectors representing the mid points $D, E$ and $F$ are $\frac{\to a + \to b}{2}, \frac{\to a + \to c}{2}$ and $\frac{\to b + \to c}{2}$ respectively.
$- - \to B E = \frac{\to a - 2 \to b + \to c}{2}$
$- - \to A F = \frac{\to b - 2 \to a + \to c}{2}$
$- - \to C D = \frac{\to a + \to b - 2 \to c}{2}$
$- - \to B E + - - \to A F = \frac{2 \to c - \to a - \to b}{2} = - - - \to C D = - - \to D C$

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The Mid-Point Theorem

Standard IX Mathematics

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