Question

A sector of circle centre O Containing an angle Φ° .Prove that perimeter of the shaded region is r(tanΦ +secΦ + π Φ/180 -1 ) ° and area of the shaded region is r^2 /2 (tanΦ- π Φ/180)

Answer 1

Hey there!


Perimeter of the shaded region
= arc AC + AB + BC

Arc AC = Φ * 2πr / 360
= Φ π r /180

Tan Φ = AB /OA
= AB /r

AB = r tan Φ

Similarly,

BC = OB -OC
= rsec Φ -r
= r(secΦ-1)

☞Perimeter = arc AC+AB+AC

= Φπr/180+ rtanΦ + rsecΦ-r

= r(tanΦ +sec Φ +π Φ/180-1)

☆Hence, your first part is proved !


☞ Area :
= Area of right triangle OAB - Area of
sector

= 1/2 *OA *AB - Φ *πr^2 /360

= r^2tanΦ/2 - πr^2 Φ /360

= r^2 /2 ( tan Φ -π Φ/180)


Hence, proved!


#hope it helps!

Answer 2

I think u got it
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thank u