In two concentric circles, the radii \(OA = r\) cm and \(OQ = 6\) cm, as shown in the figure. Chord $CD$ of the larger circle is a tangent to the smaller circle at $Q$. $PA$ is tangent to the larger circle. If $PA$ = $16$ cm and $OP = 20$ cm, find the length of $CD$.
Solution and Explanation
Given \(PA\) is tangent to the larger circle and \(OP\) is the distance from the center to the point of tangency, the Pythagorean theorem is applied to determine the radius of the larger circle.
The known values are:
\[ OP^2 = PA^2 + OA^2 \]
Substituting these values yields:
\[ 20^2 = 16^2 + r^2 \implies 400 = 256 + r^2 \implies r^2 = 144 \implies r = 12 \, \text{cm} \]
Therefore, the radius of the larger circle is \(12 \, \text{cm}\). The formula for the length of the chord is then used:
\[ CD = 2\sqrt{OP^2 - OQ^2} \]
Substituting the values into the chord length formula gives:
\[ CD = 2\sqrt{20^2 - 6^2} = 2\sqrt{400 - 36} = 2\sqrt{364} = 2 \times 19.08 = 38.16 \, \text{cm} \]
Consequently, the length of chord \(CD\) is approximately \(38.16 \, \text{cm}\).
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