Answer :

Let R = radius of larger circle in all figures.

If r = radius of circle

Circumference =

∴ Perimeter of large circle = …(eq)1

In first figure,

Diameter of larger circle = 2 × Diameter of smaller circle(as seen from figure)

Hence,

Diameter of smaller circle =

∴ Radius of smaller circle = (Radius = )

Let r = Radius of smaller circle

∴ r =

Perimeter of a smaller circle = 2 × π × r

= 2 × π × = π × R

Perimeter of 2 smaller circles = 2 × (π × R) = …(eq)1

Hence, proved.

In second figure,

Diameter of larger circle = 3 × Diameter of smaller circle(as seen from figure)

Hence,

Diameter of smaller circle =

∴ Radius of smaller circle = (Radius = )

Let r = Radius of smaller circle

∴ r =

Perimeter of a smaller circle = 2 × π × r

= 2 × π × R

Perimeter of 3 smaller circles = 3 × ( × π × R) = …(eq)1

Hence, proved.

In 3^{rd} figure,

There are 3 small circles.

Right half has radius =

Perimeter of right half = 2 × π × = π × R …(eq)2

The left half’s radius is divided in the ratio 1:2

Hence, radius of smallest circle =

Hence, radius of middle circle =

Perimeter of smallest circle = 2 × π × …(eq)3

Perimeter of middle circle = …(eq)4

Sum of perimeter of all circles = (eq)2 + (eq)3 + (eq)4

= (π × R) + (π × ) + (2 × π × )

= π × R …(same as (eq)1)

Hence, proved.

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