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
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
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.