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Out of the two concentric circles, the radius of the outer circle is $5 cm$ and the chord $AB$ of length $8 cm$ is a tangent to the inner circle. Find the radius of the inner circle.

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2 cm

3 cm

5 cm

4 cm

answer is B.

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Detailed Solution

Given, AB= 8cm, OA=OB=5cm.
Drawing the required figure,
Question Image

Perpendicular from the center to the chord bisects the chord,
Thus, AT=TB=4cm as AB = 8cm.
Also,

∠ ATO = ∠ BTO = 90 °

(Perpendicular)
In

∆ ATO

,
Applying Pythagoras theorem,

A O^{2} = A T^{2} + O T^{2}

\Rightarrow O T^{2} = A O^{2} - A T^{2}

Substituting AT=4cm and OA=5cm in the above equation,

O T^{2} = {(5 cm)}^{2} - {(4 cm)}^{2}

\Rightarrow O T^{2} = 25 {cm}^{2} - 16 {cm}^{2}

\Rightarrow O T^{2} = 9 {cm}^{2}

\Rightarrow OT = 3 cm

Thus, radius of inner circle is 3cm.
Therefore, option (2) is correct.

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