G is the centroid of triangle ABC and A₁ and B₁ are the midpoint of sides AB and AC, respectively. If $Δ_{1}$ be the area of quadrilateral GA₁AB₁ ,and $Δ$ be the area of triangle ABC ,then $\frac{Δ}{Δ_{1}}$ ,is equal to

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$3$

none of these

$\frac{3}{2}$

$\frac{1}{3}$

answer is B.

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Detailed Solution

Let pv of A,B,C are $\bar{o}, \bar{b}, \bar{c}$

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$\begin{array}{l} G = \frac{\bar{b} + \bar{c}}{3} A_{1} = \frac{\bar{b}}{2} B_{1} = \frac{\bar{c}}{2} \\ {△_{A B_{1}}}_{G} = \frac{1}{2} |A \bar{G} \times A {\bar{B}}_{1}| = \frac{1}{2} |\frac{\bar{b} + \bar{c}}{3} \times \frac{\bar{c}}{2}| = \frac{1}{12} |\bar{b} \times \bar{c}| \\ △_{A A_{1} G} = \frac{1}{2} |A \bar{G} \times A {\bar{A}}_{1}| = \frac{1}{2} |\frac{\bar{b + \bar{c}}}{3} \times \frac{\bar{b}}{12}| = \frac{1}{12} |\bar{b} \times \bar{c}| \\ △_{G A_{1} A B_{1}} = \frac{1}{6} |\bar{b} \times \bar{c}| = \frac{1}{3} . \frac{1}{2} |\bar{b} \times \bar{c}| = \frac{1}{3} △_{A B C} \end{array}$