Question

Question No. 1 If the radius of the sector in the image is 12 cm, what is the area of the violet segment (in cm)? 120° A B​

Answer 1

AsectorOABhasitsradiusr=OA=OB=12cm

andthecentralangleAOB=60

0

.

Tofindout−

ar.segmentAPB=?

Solution−

InΔABCwehaveOA=OB(radiiofthesamecircle).

∴∠OAB=∠OBA⟹∠OAB+∠OBA=2∠OAB.

Now∠AOB+∠OAB+∠OBA=60

0

+2∠OAB=180

O

(anglesumpropertyoftriangles).

⟹2∠OAB=120

O

⟹∠OAB=∠OBA=60

0

.

∴ΔAOBisanequilateralonewithsidea=12cm.

Soar.ΔAOB=

4

3

a

2

=

4

3

12

2

cm

2

=36

3

cm

2

.

Againar.sector=

360

o

θ

×πr

2

whenθisthecentralangle

&ristheradiusofthesector.

Herer=a=12cmandθ=60

0

.

ar.sectorAOBP=

360

o

60

o

×3.14×12

2

cm

2

=75.36cm

2

.

ar.segmentAPB=ar.sectorAOBP−ar.ΔAOB

=(75.36−36

3

)cm