Question

In fig., are shown two arcs PAQ and PBQ . Arc PAQ is a part of circle with centre O and radius OP while arc PBQ is a semi-circle drawn on PQ as diameter with centre M. If OP=PQ=10 cm , show that area of shaded region is 25 ( ?3 -?/6 ) cm?

Answer 1

Answer:

Step-by-step explanation:

Find the area of the semi circle PBQ

Area of a semi circle = 1/2 πr²

Area of a semi circle = 1/2 π(5)² = 25π/2

Find the area of the triangle:

OP = 10 cm

OQ = 10 cm (radius)

PQ = 5 + 5 = 10 cm

Since all the length are equal, it is an equilateral triangle

Area of an an equilateral triangle = √3/4 (side)²

Area = √3/4 (10)²

Area = 25√3

Find the area of the sector OPQ:

It is an equilateral triangle

⇒∠POQ = 60º

Area of the sector = 60/360 x π(10)²

Area = 50π/3

Find the area of the segment PAQ:

Area of the segment = Area of the sector - Area of the triangle

Area = 50π/3  - 25√3

Area = 25(2π/3  - √3)

Find the area of the shaded region:

Area = 25π/2 -  25(2π/3  - √3)

Area = 25(π/2 - 2π/3 + √3)

Area = 25( √3 - π/6)

Thanks!!

Answer 2

$\underline{25( \sqrt{3} - \frac{\pi}{6} )}$