In the given figure, PQRS is a rectangle of length 10 2 cm and breadth 5 2 cm . If PEQ is an isosceles triangle inscribed in the semi-circle with diameter PQ, then find the area of the shaded region.

A
$2003 cm 2$
B
$2007 cm 2$
C
$2005 cm 2$
D
$2009 cm 2$

Solution:

Given rectangle PQRS of length

102 cm

and breadth

52 cm

Area of the quadrant

= π r 2 4 c m 2

(r is for radius)
Area of the square

A = (s i d e) 2 cm 2

.
OE is parallel to PS and QR in such a way that it splits into two squares with sides

52 c m

.
Area of the squares POES and QOER

(A) = (s i d e) 2 c m 2

⇒ A = (52) 2 c m 2 ⇒ A = 50 c m 2

Area of triangle POE and QOE

= A r e a o f s q u a r e 2

⇒ A r e a o f t r i a n g l e P O E a n d Q O E = 502 ⇒ A r e a o f t r i a n g l e P O E a n d Q O E = 25 cm 2

Area of the quadrant OPE and QOE with equal radius (square's side-to-side),

⇒ A q u a d r a n t = π r 2 4 c m 2 ⇒ A q u a d r a n t = π (52) 2 4 c m 2 ⇒ A q u a d r a n t = 252 π c m 2

Area of shaded region= Areas of quadrant OPE and QOE with equal radius (equal to side of square)- Areas of triangle OPE and OQE
Area of the shaded region:

⇒ A s h a d e d r e g i o n = 2 × 252 π − 2 × 25 cm 2 ⇒ A s h a d e d r e g i o n = 2 × 252 × 227 − 2 × 25 cm 2 ⇒ A s h a d e d r e g i o n = 2007 cm 2

Hence, the area of the shaded region is

2007 cm 2

.
Therefore, the correct option is 2.

In the given figure, PQRS is a rectangle of length $102 cm$ and breadth $52 cm$ . If PEQ is an isosceles triangle inscribed in the semi-circle with diameter PQ, then find the area of the shaded region.

Solution:

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