Question

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Bonjour°==

Solve This Please (⊙_◎)
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Answer 1

First, find the lengths of the region for the perimeter.

Perimeter = AB + PB + Arc AP.

First, find the Arc length AP.

AP =

$\frac{ \theta }{360} \times 2\pi \: r = \frac{ \theta \: \pi \: r}{180}$

Now, in right angled triangle OAB,

$\tan( \theta ) = \frac{ab}{r}$

AB = r tan{\theta}

$\sec( \theta ) = \frac{ob}{r}$

OB = r sec {\theta }

OB = OP + PB.

$</p><p>{\underline {\therefore {\red{PB \: = \: r sec ( \theta ) - r }}}}</p><p>$

Now substituting the values, we get

Perimeter =

$r \: \tan( \theta ) + r \: \sec( \theta) - \: r \: + \frac{\pi. \theta .r }{180}$

$= r . \: ( \tan( \theta ) + \sec( \theta) + \frac{\pi \theta}{180} - 1)$

$</p><p>\huge{\fbox{\fbox{\bigstar{\mathfrak{\blue{Hence \: Proved }}}}}}$