SATHEE: Properties of Triangles 2 Question 8

Properties of Triangles 2 Question 8

8. Prove that a $△ A B C$ is equilateral if and only if $\tan A + \tan B + \tan C = 3 \sqrt{3}$ .

(1998, 8M)

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Solution:

If the triangle is equilateral, then

$A = B = C = 60^{\circ}$

$\Rightarrow \tan A + \tan B + \tan C = 3 \tan 60^{\circ} = 3 \sqrt{3}$

Conversely assume that,

$\tan A + \tan B + \tan C = 3 \sqrt{3}$

But in $△ A B C, A + B = 180^{\circ} - C$

Taking tan on both sides, we get

$\tan (A + B) = \tan (180^{\circ} - C)$

$\Rightarrow \frac{\tan A + \tan B}{1 - \tan A \tan B} = - \tan C$

$\Rightarrow \tan A + \tan B = - \tan C + \tan A \tan B \tan C$

$\Rightarrow \tan A + \tan B + \tan C = \tan A \tan B \tan C = 3 \sqrt{3}$

$\Rightarrow$ None of the $\tan A, \tan B, \tan C$ can be negative

So, $△ A B C$ cannot be obtuse angle triangle.

Also, $A M \geq G M$

$\Rightarrow \frac{1}{3} [\tan A + \tan B + \tan C] \geq [\tan A \tan B \tan C]^{1 / 3}$

$\Rightarrow \frac{1}{3} (3 \sqrt{3}) \geq (3 \sqrt{3})^{1 / 3} \Rightarrow \sqrt{3} \geq \sqrt{3}$ .

So, equality can hold if and only if $\tan A = \tan B = \tan C$

or $A = B = C$ or when the triangle is equilateral.

Properties of Triangles 2 Question 8

8. Prove that a $△ A B C$ is equilateral if and only if $\tan A + \tan B + \tan C = 3 \sqrt{3}$ .

Solution:

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Properties of Triangles 2 Question 8

8. Prove that a △ABC is equilateral if and only if tan⁡A+tan⁡B+tan⁡C=33.

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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8. Prove that a $△ A B C$ is equilateral if and only if $\tan A + \tan B + \tan C = 3 \sqrt{3}$ .