Question

3) A circle is inscribed in an equilateral triangle of side
20 cm, touching its sides. What is the area of the circle?
A2​

Answer 1

Answer:

A circle is inscribed in an equilateral triangle ABC with side 12 cm, touching its sides. Find the radius of the inscribed circle and the area of the shaded part.

973584

Medium

Answer

⇒ Here, AB=BC=AC=12cm

⇒ Let OP=OR=OQ=r

⇒ We have O as the incenter and OP,OQ and OR are equal.

⇒ ar(△ABC)=ar(△OAB)+ar(△OBC)+ar(△OCA)

4

3

×(side)

2

=(

2

1

×OP×AB)+(

2

1

×OQ×BC)+(

2

1

×OR×AC)

⇒

4

3

×(12)

2

=(

2

1

×r×12)+(

2

1

×r×12)+(

2

1

×r×12)

⇒

4

3

×(12)

2

=3(

2

1

×12×r)

∴ r=

18

36

3

∴ r=2

3

cm

⇒ Area of the shaded region = Area of △ABC - Area of circle.

⇒ Area of the shaded region =

4

3

×(12)

2

−

7

22

×(2

3

)

2

⇒ Area of the shaded region =(62.35−37.71)cm

2

=24.64cm

2

solution