Question

Let $H$ be the orthocenter of $\triangle ABC$ where $A \equiv(0,0)$, $B \equiv(8,0)$ and $C \equiv(4,4 \sqrt{3}) .$ If $G _{1}, G _{2}, G _{3}, G _{4}, \ldots .$ are centroids of triangles $\Delta AHB , \Delta AG _{1} B , \Delta AG _{2} B , \Delta AG _{3} B , \ldots . .$ respectively then $\displaystyle \lim _{n \rightarrow \infty} \displaystyle\sum_{i=1}^{n} \text{Ar}\left(\Delta A G_{i} B\right)$ equals

• A. $8 \sqrt{3}$ sq. units
• B. $16 \sqrt{3}$ sq. units
• C. $\frac{8}{\sqrt{3}}$ sq. units
• D. $\frac{16}{\sqrt{3}}$ sq. units

Answer 1

Answer: (C)

$\triangle ABC$ is equilateral
$H \rightarrow$ orthocenter or centroid
$H =\left(4, \frac{4}{\sqrt{3}}\right) ; G _{1}=\left(4, \frac{4}{3 \sqrt{3}}\right) ; G _{2}=\left(4, \frac{4}{9 \sqrt{3}}\right)$
$: \quad \quad :$
$: \quad \quad :$
$G_{n}=\left(4, \frac{4}{3 \sqrt[n]{3}}\right)$
$\therefore \displaystyle\lim _{n \rightarrow \infty}\left(A_{1}+A_{2}+\ldots A_{n}\right) $
$ \left[\right.$ All $\Delta'$ s are isosceles $]$
$=\frac{1}{2} \times 8\left[\frac{4}{3 \sqrt{3}}+\frac{4}{9 \sqrt{3}}+\frac{4}{3 \sqrt[n]{3}}\right]$
$=4 \times \frac{4}{3 \times \sqrt{3}} \displaystyle\lim _{n \rightarrow \infty}\left[1+\frac{1}{3}+\frac{1}{3^{2}}+\ldots .+\frac{1}{3^{n-1}}\right]$
$=\frac{16}{3 \sqrt{3}} \times \frac{1}{1-\frac{1}{3}}=\frac{8}{\sqrt{3}}$