Question

For a triangle $ABC$, the value of $\cos 2A+\cos 2B+\cos 2C$ is least. If its inradius is $3$ and incentre is $M$, then which of the following is NOT correct?

• A. area of $△ABC$ is $\frac{27\sqrt{3}}{2}$
• B. $\sin 2A+\sin 2B+\sin 2C=\sin A+\sin B+\sin C$
• C. perimeter of $△ABC$ is $18\sqrt{3}$
• D. $\overrightarrow{MA}\cdot \overrightarrow{MB}=-18$

Answer 1

Answer: (A)

If $\cos 2A+\cos 2B+\cos 2C$ is minimum then $A=$ $B=C=60^{\circ }$
So $△ABC$ is equilateral
Now in-radias $r=3$
So in $△MBD$ we have
$\mathrm{Tan}30^{\circ }=\frac{MD}{BD}=\frac{r}{a/2}=\frac{6}{a}$
$1/\sqrt{3}=\frac{1}{a}=a=6\sqrt{3}$
Perimeter of $△ABC=18\sqrt{3}$
Area of $△ABC=\frac{\sqrt{3}}{4}{a}^2=27\sqrt{3}$