First slide

Cross product of vectors or vector product of vectors

Question

G is the centroid of triangle ABC and A₁ and .B₁ are the midpoints of sides AB and AC, respectively. If $∆_{1}$ is the area of quadrilateral GA₁,AB₁ and $∆$ is the area of triangle ABC then $\frac{∆_{1}}{∆_{2}}$ is equal to

Moderate

A

$\frac{3}{2}$
B

3
C

$\frac{1}{3}$
D

none of these

Solution

Let P.V. of A,B and C be $\vec{0}, \vec{b} and \vec{c}$ respectively
Therefore,
$\begin{array}{l} \vec{G} = \frac{\vec{b} + \vec{c}}{3} \\ {\vec{A}}_{1} = \frac{\vec{b}}{2}, {\vec{B}}_{1} = \frac{\vec{c}}{2} \\ Δ_{{AB}_{1} G} = \frac{1}{2} |\vec{AG} \times \vec{{AB}_{1}}| \\ = \frac{1}{2} |\frac{\vec{b} + \vec{c}}{3} \times (\frac{\vec{c}}{2})| \\ = \frac{1}{12} | \vec{b} \times \vec{c} | \\ Δ_{{AA}_{1} G} = \frac{1}{2} |\vec{AG} \times \vec{{AA}_{1}}| = \frac{1}{2} |\frac{\vec{b} + \vec{c}}{3} \times (\frac{\vec{b}}{2})| \\ = \frac{1}{12} | \vec{b} \times \vec{c} | \end{array}$
$\begin{array}{l} \Rightarrow Δ_{{GA}_{1} {AB}_{1}} = \frac{1}{6} | \vec{b} \times \vec{c} | = \frac{1}{3} \cdot \frac{1}{2} | \vec{b} \times \vec{c} | \\ = \frac{1}{3} Δ_{ABC} \\ \Rightarrow \frac{Δ}{Δ_{1}} = 3 \end{array}$

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