Question

find the area of segment AYB if the radius is 21cm
pls ans fast.....

Answer 1

Question :

From the figure in attachment , find the area of segment AYB if radius is 21 cm .

Answer:

$\huge{\boxed{\frac{( 1848- 441\sqrt{3})}{4}cm^2}}$

Step-by-step explanation:

Figure in attachment .

Area of sector OAYB

Area = angle / 360° × π r²

⇒ Area = 120° / 360° × π r²

⇒ Area = π r² / 3

⇒ Area = 22/7 × 21 × 21/3

⇒ Area = 22/7 × 21 × 7

⇒ Area = 22 × 21

⇒ Area = 462 cm²

Area of Δ OAB

Area = 1/2 × b × h

OH is constructed so that OH = h .

In Δ OAH and Δ OHB ,

∠AHO = ∠BHO [ 90 ]

OH = OH [ common ]

AO = OB [ radius ]

Δ OAH ≅ Δ OHB [ R.H.S ]

∴ AH = HB [ c.p.c.t ]

cos 30 = BH/OB

⇒ √3/2 = BH/21

⇒ BH = 21 × √3/2

b = BH + AH

⇒ b = 21 × √3/2 + 21 × √3/2

⇒ b = 2 × 21 × √3/2

⇒ b = 21√3

By Pythagoras theorem :

h² = OB² - BH²

⇒ h² = 21² - ( 21 × √3/2 )²

⇒ h² = 441 - 441 × 3/4

⇒ h² = 441 ( 1 - 3/4 )

⇒ h² = 441 ( 1/4 )

⇒ h² = 441/4

⇒ h = 21/2

Area = 1/2 × b × h

⇒ 1/2 × 21√3 × 21/2

⇒ 441√3/4

Area of segment

Area = 462 - 441√3/4

⇒ ( 1848 - 441√3 )/4

The area of segment is ( 1848 - 441√3 )/4  cm² .