Question

If median $AD$ of a triangle $ABC$ makes an angle $\frac{\pi }{6}$ with side $BC$, then the value of $(\cot B-\cot C{)}^2$ is equal to:

• A. $9$
• B. $12$
• C. $18$
• D. $24$

Answer 1

The correct option is B $12$

As $AD$ is the median
So, $BD=DC$
Applying $m-n$ theorem
$(BD+DC)\cot \frac{\pi }{6}=DC\cot B-BD\cot C$
$(BD+BD)\cot \frac{\pi }{6}=BD\cot B-BD\cot C$
$2\sqrt{3}=\cot B-\cot C$
Taking square of both sides
$4\times 3=(\cot B-\cot C{)}^2$
$\therefore (\cot B-\cot C{)}^2=12$