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The sum of the radii of inscribed and circumscribed circles for an $n$ sides regular polygon of side $a$ , is

Option 1)
$\frac{a}{2}cot\left ( \frac{\pi }{2n} \right )\; \;$

Option 2)
$\; a\, cot\left ( \frac{\pi }{2n} \right )\; \;$

Option 3)
$\; \frac{a}{4}cot\left ( \frac{\pi }{2n} \right )\; \;$

Option 4)
$\; a\, cot\left ( \frac{\pi }{n} \right )$

Answers (1)

best_answer

As we learnt in

Trigonometric Ratios of Functions -

$\sin \Theta = \frac{Opp}{Hyp}$

$\cos \Theta = \frac{Base}{Hyp}$

$\tan \Theta = \frac{Opp}{Base}$

- wherein

Trigonometric Ratios of Functions

let AB represent any side of polygon with O as center AB=a

$\angle AOB=\frac{2\pi}{n}$

$\angle COB=\frac{\pi}{n}$

OC = in radius, OA=OB = circum radius = R

Now, $In\ \Delta OCB, \cot \angle COB=\frac{r}{\frac{a}{2}}\Rightarrow r=\frac{a}{2}\cot \frac{\pi}{n}$

Similarly, $In\ \Delta OCB, \sin \angle COB={\frac{\frac{a}{2}}{R}}\Rightarrow R=\frac{a}{2}\: \frac{1}{\sin \frac{\pi}{n}}$

$Thus\ R+r = \frac{a}{2}[\cot \frac{\pi}{n}+\frac{1}{\sin \frac{\pi}{2}}]$

$= \frac{a}{2}\ sin \frac{\pi}{n}[cos \frac{\pi}{n}+1]$

$= \frac{a}{2}\ \frac{2\cos ^{2}\frac{\pi}{2n}}{2\sin\frac{\pi}{2n}\cos\frac{\pi}{2n} }$

$= \frac{a}{2}\cot\frac{\pi}{2n}$

Option 1)

$\frac{a}{2}cot\left ( \frac{\pi }{2n} \right )\; \;$

This is correct option

Option 2)

$\; a\, cot\left ( \frac{\pi }{2n} \right )\; \;$

This is incorrect option

Option 3)

$\; \frac{a}{4}cot\left ( \frac{\pi }{2n} \right )\; \;$

This is incorrect option

Option 4)

$\; a\, cot\left ( \frac{\pi }{n} \right )$

This is incorrect option

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