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