Question

A chord PQ of a circle with radius r,subtends an angle θ at the centre.Show that the area of the minor segment PQ is 1/2 r^2 (θ-sin⁡θ ),and write down the area of the major segment PQ in terms of r and θ.

Answer 1

Step-by-step explanation:

of minor segment=

8

1

×πr

2

Area of sector-area of △AOB=

8

πr

2

In △AOB,OM⊥AB,∴∠AOM=∠MOB=

2

θ

and AM=MB=

2

AB

In △AOM,sin

2

θ

=

OA

AM

=

r

AM

⇒AM=rsin

2

θ

cos

2

θ

=

OA

OM

=

r

OM

⇒OM=rcos

2

θ

∴ar△AOM=

2

1

×OM×AM=

2

1

×rcos

2

θ

×sin

2

θ

=

2

r

2

sin

2

θ

cos

2

θ

Similarly, ar△BOM=

2

r

2

sin

2

θ

cos

2

θ

∴ar△AOB=ar△AOM+ar△BOM=r

2

sin

2

θ

cos

2

θ

⇒

360°

θ

×πr

2

−r

2

sin

2

θ

cos

2

θ

=

8

πr

2

⇒r

2

sin

2

θ

cos

2

θ

+

8

πr

2

=

360°

θ

×πr

2

⇒

8

r

2

=

360

πθ

×r

2

⇒8sin

2

θ

cos

2

θ

+π=

45

πθ