Properties of Triangles Question 1
Question 1 - 01 February - Shift 1
For a triangle $ABC$, 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?
(1) Perimeter of $\triangle ABC$ is $18 \sqrt{3}$
(2) $\sin 2 A+\sin 2 B+\sin 2 C=\sin A+\sin B+\sin C$
(3) $\overrightarrow{{}MA} \cdot \overrightarrow{{}MB}=-18$
(4) area of $\triangle ABC$ is $\frac{27 \sqrt{3}}{2}$
Show Answer
Answer: (4)
Solution:
If $\cos 2 A+\cos 2 B+\cos 2 C$ is minimum then $A=B=C=60^{\circ}$
So $\triangle ABC$ is equilateral
Now inradias $r=3$
So in $\triangle MBD$ we have
$tan 30^{\circ}=\frac{M D}{B D}=\frac{r}{a / 2}=\frac{6}{a}$
$\frac{1} {\sqrt{3}}=\frac{1}{a}=a=6 \sqrt{3}$
Perimeter of $\triangle ABC=18 \sqrt{3}$
Area of $\triangle ABC=\frac{\sqrt{3}}{4} a^{2}=27 \sqrt{3}$
