out of the two concentric circle, the radius of the outer circle is $5$ cm and the chord AC of length $8$ cm is a tangent to the inner circle, then find the radius of the inner circle.

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Solution

in the two concentric circle
$A B$ is the chord of bigger circle and tangent to the smaller circle
$A B = 8 c m$
$O A = 5 c m$
$O C ⊥ A B$
using Pythagoras theorem
$\begin{matrix} {(A O)}^{2} = {(O C)}^{2} + {(C A)}^{2} {(5)}^{2} = {(O C)}^{2} + (\frac{A C}{2}) L e t O C = x x = \sqrt{25 - 16} x = 3 \end{matrix}$
Radius of inner circle is $3 c m$

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