Question

$\sf \pink{ maths \: legends....!!!}$
$\huge \bf \: {\underline{\underline{\red{ ✿Question}}}}$
The attached figure shows a sector OAP of a circle with centre O, containing ∠$\theta$. AB is the perpendicular to the radius OA and meets OP produced at B. Prove that the perimeter of shaded region is :-

$\sf \purple{ r \bigg[\tan \theta + \sec \theta + \dfrac{ \pi \theta}{180} - 1 \bigg]}$
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Answer 1

$\\ \large \underline { \underline \pink{\rm☯\: Question :- }}$

The attached figure shows a sector OAP of a circle with centre O, containing ∠$\theta$. AB is the perpendicular to the radius OA and meets OP produced at B. Prove that the perimeter of shaded region is :-

$\sf \orange{ r \bigg[\tan \theta + \sec \theta + \dfrac{ \pi \theta}{180} - 1 \bigg]}$

$\\ \large \underline { \underline \pink{\rm☯\: Solution :- }}$

Perimeter of shaded region $\\ = \sf \: AB + PB + arc \: length \: AP \: --- \: \: \: \red{ (i)}$

$\\ \sf \: \: \: \: \: \: \: Arc \: length \: AP = \frac{ \theta}{360} \times 2\pi r$

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \frac{ \pi\theta r}{360} - - - - - - \orange{(ii)}$

Now, In right angled ∆OAB,

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: tan \: \theta \: = \dfrac{AB}{r} \:$

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \therefore \: \: \: \: \: \: {AB} = {r} \: tan \: \theta \: \: - - - - (iii)$

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: Sec \: \theta \: = \dfrac{OB}{r} \:$

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \therefore \: \: \: \: \: \: {OB} = {r} \: sec \: \theta \:$

$\ \\ \sf \: OB = OP + PB$

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \therefore \: \: \: \: \: \: {PB} = {r} \: sec \: \theta \: - \: r ----(iv)$

Now, substitute (ii) ,(iii) ,(iv) in equation (i), then

we get

$\\ \: \: \: \: \: \: \: \: \implies \: \sf { \sf {r \: tan \theta +r \: sec \theta - r + \dfrac{ \pi \theta}{180} - 1}}$

$\\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{\boxed{ \: \sf \red{ r \bigg[\tan \theta + \sec \theta + \dfrac{ \pi \theta}{180} - 1 \bigg]}}}$