Question

Arjav was drawing a triangle .He drew line segment PQ and two arcs as shown below .The radii of the circles containing the arcs are 6cm and 4cm and their centers are P and Q respectively .Which of the following could be the length of PQ ​

Answer 1

PQ and AB are the arcs of two concentric circles of radii 7 cm and 3.5 cm respectively.

Let r

1

and r

2

be the radii of the outer and the inner circle respectively.

Suppose θ be the angle subtended by the arcs at the centre O.

Then r

1

=7 cm, r

2

=3.5 cm and θ=30

o

Area of the shaded region = Area of sector OPQ − Area of sector OAB

=

360

o

θ

πr

1

2

−

360

o

θ

πr

2

2

=

360

o

θ

π(r

1

2

−r

2

2

)

=

360

o

30

o

×

7

22

[7

2

−3.5

2

]

=

12

1

×

7

22

×(49−12.25)

=

12

1

×

7

22

×36.75

=9.625

Therefore, the area of the shaded region is 9.625 cm

2

.