Question

Let $\overrightarrow{a}=\overrightarrow{BC},\overrightarrow{b}=\overrightarrow{CA},\overrightarrow{c}=\overrightarrow{AB}$ be the sides of the triangle $ABC$. If $G$ is the centroid of $\Delta ABC$ such that $\overrightarrow{GB}$ and $\overrightarrow{GC}$ are inclined at an obtuse angle, then

• A. $5a^2>b^2+c^2$
• B. $5c^2>a^2+b^2$
• C. $5b^2>a^2+c^2$
• D. None of these

Answer 1

The correct option is A $5a^2>b^2+c^2$

Let $\overline{A}=0,\overline{AB}=\overline{b},\overline{AC}=\overline{c}$

Then $\overline{AG}=\frac{\overline{b}+\overline{c}}{3}$
Obtuse angle implies $\overline{GB}.\overline{GC}<0.$
$\Rightarrow 5(\overline{b}.\overline{c})-2|b{|}^2-2|c{|}^2<0.$

$\overline{b}.\overline{c}=\frac{c^2+b^2-a^2}{2}\Rightarrow 5a^2>b^2+c^2$