Question

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 $CD$.

Answer 1

Since \(PA\) is tangent to the larger circle and \(OP\) is the distance from the center to the point of tangency, we can use the Pythagorean theorem to find the radius of the larger circle.

We already know:

\[ OP^2 = PA^2 + OA^2 \]

Substituting the values:

\[ 20^2 = 16^2 + r^2 \implies 400 = 256 + r^2 \implies r^2 = 144 \implies r = 12 \, \text{cm} \]

Thus, the radius of the larger circle is \(12 \, \text{cm}\), and we use the formula for the length of the chord:

\[ CD = 2\sqrt{OP^2 - OQ^2} \]

Substitute the values:

\[ CD = 2\sqrt{20^2 - 6^2} = 2\sqrt{400 - 36} = 2\sqrt{364} = 2 \times 19.08 = 38.16 \, \text{cm} \]

Thus, the length of chord \(CD\) is approximately \(38.16 \, \text{cm}\).