Three circles of radius 2 cm touch one another externally These circles are circumscribed by a circle of radius - Maths - Areas Related to Circles

Question

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

Lalit Mehra · Accepted Answer

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,

$\cos 30 ° = \frac{BD}{OB} ∴ \frac{\sqrt{3}}{2} = \frac{2 cm}{(x - 2) cm} \Rightarrow \sqrt{3} x - 2 \sqrt{3} = 4 \Rightarrow \sqrt{3} x = 2 \sqrt{3} + 4 \Rightarrow x = \frac{2 \sqrt{3} + 4}{\sqrt{3}} = 2 + \frac{4}{\sqrt{3}}$

Thus, the value of x is $(2 + \frac{4}{\sqrt{3}}) cm$ .

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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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