Question

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

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

Answer 1

Answer: (A)

area of $\triangle A B C$ is $\frac{27 \sqrt{3}}{2}$

Answer 2

If cos2A+cos2B+cos2C is minimum then A= B=C=60∘
So △ABC is equilateral
Now in-radias r=3
So in △MBD we have
Tan30∘=BDMD​=a/2r​=a6​
1/3​=a1​=a=63​
Perimeter of △ABC=183​
Area of △ABC=43​​a2=273​