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  • If A, B, and C is the angles of a triangle, and <img alt="\tan A + \tan B+\tan C=3\sqrt{3}" src="http://entrancecorner.oncodecogs.com/gif.latex?%5Ctan%20A%20&plus;%20%5Ctan%20B&pl

If A, B, and C is the angles of a triangle, and \tan A + \tan B+\tan C=3\sqrt{3}, then which of the following is true ?

Option: 1

Three different triplet of angles are possible  


Option: 2

ABC is a right triangle  


Option: 3

ABC can be equilateral triangle but not necessarily 


Option: 4

ABC is a equilateral triangle 


Answers (1)

best_answer

\text{Given }\ \ \ \tan A + \tan B+\tan C=3\sqrt{3}\\ \text{ we know }\\ \tan A+\tan B+\tan C=\tan A \tan B \tan C \ \ \ \{ \text{ for triangle ABC} \}\\ A.M. \geq G.M\\ \frac{\tan A+\tan B+\tan C}{3}\geq (\tan A \tan B \tan C)^\frac{1}{3}\\ \sqrt{3}\geq (3\sqrt{3})^\frac{1}{3}\\ A.M.=G.M. \ \ \text{only possible when } {tan A=tan B=tan C} \text{ or A=B=C}\\ A+B+C=\pi\\ so\ A=B=C=\frac{\pi}{3}

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Riya

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