Question

If in a triangle $ABC,\tan A+\tan B+\tan C=6$ and $\tan A\tan B=2$, then the triangle is:

• A. right angled
• B. obtuse angled
• C. acute angled
• D. isosceles

Answer 1

The correct option is C acute angled

$A+B+C=180A+B=180-C\tan (A+B)=-\tan C\frac{(\tan A+\tan B)}{(1-\tan A\tan B)}=-\tan C\tan A+\tan B+\tan C=\tan A\tan B\tan C=6$
Now, $\tan C=3.$ So, the angle $C$ is acute.

$\tan A+\tan B=3$

$\tan A.\tan B=2$

$\therefore \tan A=2,\tan B=1$

$A$ and $B$ are also acute.
Therefore, it is acute angled triangle.