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In the Figure Given Below, Pqrs is Square Lawn with Side Pq = 42 Metres. Two Circular Flower Beds Are There on the Sides Ps and Qr with Centre at O, - Mathematics

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Sum

In the figure given below, PQRS is square lawn with side PQ = 42 metres. Two circular flower beds are there on the sides PS and QR with centre at O, the intersections of its
diagonals. Find the total area of the two flower beds (shaded parts).

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Solution

Area of the square lawn PQRS = 42 m x 42 m

Let OP = OS = xm

So, x2 + x2  = (42)

⇒ 2x2 = 42 × 42

⇒ x2 = 21 × 42

Now,

area of sector POS =

`90/360 xx pi"x"^2 = 1/4 x (pi "x"^2)`  ......(i)

`= 1/4 xx 22/7 xx 21 xx 42 "m"^2`  .........(ii)

Also,

Area of Δ POS = `1/4 xx "Area of square lawn PQRS"`

`= 1/4 xx (42 xx 42)^2 "m"^2`     (∠POQ = 90°)  ...(iii)

So, 

Area of flower bed PSP = Area of sector POS - Area of Δ POS

`= 1/4 xx 22/7 xx 21 xx 42 - 1/4 xx 42 xx 42`   [from (ii) and (iii)]

`= 1/4 xx 21 xx 42 xx (22/7 - 2)`

`= 1/4 xx 21 xx 42 xx (8/7) "m"^2`

Therefore area of the two flower beds = `2 xx 1/4 xx 21 xx 42 xx (8/7)`

= 504 m2 

Hence , the total area of the flower beds 504 m2 .

Concept: Circumference of a Circle
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