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Two concentric circles, a chord of length $8 cm$ of the larger circle touches the smaller circle. If the radius of the larger circle is $5 cm$ then the radius of the smaller circle will be ____ cm.

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Detailed Solution

Two concentric circles, a chord of length

8 cm

of the larger circle touches the smaller circle. If the radius of the larger circle is

5 cm

then the radius of the smaller circle will be 3cm.
Given that a chord of the length 8cm and the radius of the larger circle is

5 cm

.
Drawing a perpendicular from the center

O

to the chord,
Let

A B

and join the

O A

and

O B

to form the radius of the bigger circle.
Question Image

Using congruency of triangles on

Δ O A D, Δ O B D

,
OA = OB [Radius]
OD = OD [Common]
DA= DB [OD is perpendicular]
Thus by RHS postulate, both the triangles are congruent and have corresponding equal sides.

A D, B D = A B 2 A D, B D = 82 A D, B D = 4 cm

Using Pythagoras theorem in a triangle

O A D

to calculate,

O A 2 = A D 2 + O D 2 52 = 42 + O D 2 O D 2 = 25 − 16 O D 2 = 9

Solving further to calculate

O D

O D = 9 O D = 3 cm

Thus, the radius of the smaller circle is

3 c m

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