Properties of Triangles 3 Question 13

13. The sides of a triangle inscribed in a given circle subtend angles $\alpha, \beta$ and $\gamma$ at the centre. The minimum value of the arithmetic mean of $\cos \alpha+\frac{\pi}{2}, \cos \beta+\frac{\pi}{2}$ and $\cos \gamma+\frac{\pi}{2}$ is ….

$(1987,2 M)$

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Answer:

Correct Answer: 13. $-\frac{\sqrt{3}}{2}$

Solution:

  1. Since, sides of a triangle subtends $\alpha, \beta$, $\gamma$ at the centre.

$\therefore$ Now, arithmetic mean

$$ =\frac{\cos \frac{\pi}{2}+\alpha+\cos \frac{\pi}{2}+\beta+\cos \frac{\pi}{2}+\gamma}{3} $$

As we know that, $A M \geq G M$, i.e.

AM is minimum, when $\frac{\pi}{2}+\alpha=\frac{\pi}{2}+\beta=\frac{\pi}{2}+\gamma$

or

$$ \alpha=\beta=\gamma=120^{\circ} $$

$\therefore$ Minimum value of arithmetic mean

$$ =\cos \frac{\pi}{2}+\alpha=\cos \left(210^{\circ}\right)=-\frac{\sqrt{3}}{2} $$



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