Properties Of Triangles Ques 46
- 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 $\cos \left( \alpha+\frac{\pi}{2} \right), \cos \left( \beta+\frac{\pi}{2} \right)$ and $\cos \left( \gamma+\frac{\pi}{2} \right)$ is ….
$(1987,2 M)$
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Answer:
Correct Answer: 46.$-\frac{\sqrt{3}}{2}$
Solution:
Since the sides of a triangle subtend angles $\alpha, \beta$, $\gamma$ at the centre.
$\therefore$ Now, arithmetic mean.
$$ =\frac{\cos \left( \frac{\pi}{2}+\alpha \right)+\cos \left( \frac{\pi}{2}+\beta \right)+\cos \left( \frac{\pi}{2}+\gamma \right)}{3} $$
As we know that, $AM \geq GM$, 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
$$ =\cos \left(\frac{\pi}{2}+\alpha\right)=\cos \left(210^{\circ}\right)=-\frac{\sqrt{3}}{2} $$
BETA
