Three circles of radius 2 cm touch one another externally. These circles are circumscribed by a circle of radius x. Find the value of x and the area of shaded region

Let A, B and C be the centres of three circles touching each other externally.

Suppose O is the centre of the circumscribed circle.

Radius of three circles = 2 cm

Radius of the circumscribed circle = x cm

BC = BD + CD = 2 cm + 2cm = 4 cm

Similarly, AB = AC = 4 cm

In ΔABC,

AB = BC = CA

∴ ΔABC is a equilateral triangle.

⇒ ∠ABC = ∠ACB = ∠BAC = 60°

ΔOBD ΔOCD  (SSS congruence criterion)

∴ ∠OBD = ∠OCD = 30°  (CPCT)

and ∠ODB = ∠ODC = 90°  (CPCT & linear pair)

OB = (x – 2) cm

In ΔOBD,

Thus, the value of x is .

 

The shaded region is not mentioned in the question. We will be able to offer you meaningful help only if you clearly mention the shaded region  in the question.

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