Question

A chord of a circle of radius 12 cm subtends an angle of 120 at the centre . Find the area of the corresponding segment of the circle. ​

Answer 1

Answer:

answr

search

What would you like to ask?

MATHS

A chord of a circle of radius 12 cm subtends an angle of 120

∘

at the centre. Find the area of the corresponding segment of the circle.(Use π=3.14 and

3

=1.73

1048948

Share

Study later

ANSWER

Given that:-

Radius of circle (r)=12cm

θ=120°

To find:-

Area of segment APB=?

Solution:-

Area of sector OAPB(A

1

)=

360°

θ

×πr

2

⇒A

1

=

360

120

×3.14×12

2

=150.72cm

2

Let M be the point on AB such that AB⊥OM

∴∠OMA=∠OMB=90°

Now in △OMA and △OMB,

∠OMA=∠OMB[ each 90°]

OA=OB[∵OA=OB=r]

OM=OM[ common ]

By R.H.S. congruency,

△OMA≅△OMB

Now by CPCT,

∠AOM=∠BOM

AM=BM

Therefore,

∠AOM=∠BOM=

2

1

∠AOB=60°

AM=BM=

2

1

AB.....(1)

Now in right angled △AOM

sin60=

OA

AM

[∵sinθ=

hypotenuse

Perpndicular

]

⇒sin60=

12

AM

⇒AM=6

3

cm

cos60=

OA

OM

[∵cosθ=

hypotenuse

base

]

⇒cos60=

12

OM

⇒OM=6cm

Now, from eq

n

(1), we have

AB=2AM=12

3

cm

Now, area of △AOB(A

2

) will be-

A

2

=

2

1

×AB×OM=

2

1

×12

3

×6=36

3

cm

2

=62.28cm

∴ Area of segment APB=A

1

−A

2

=150.72−62.28=88.44cm

2

Hence the aea of the corresponding segment of the circle is 88.44cm

2

.

solution