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 $38.16 \, \text{cm}$.