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

Step 1: Using the Pythagorean theorem to find the radius of the larger circle:

We are given that \( PA \) is tangent to the larger circle, and \( OP \) is the distance from the center \( O \) to the point of tangency \( P \). Since \( PA \) is tangent to the circle at \( A \), we can apply the Pythagorean theorem in the right triangle \( OPA \), where:
- \( OP \) is the distance from the center to the point of tangency (which we will find),
- \( PA \) is the length of the tangent,
- \( OA \) is the radius of the smaller circle.
The Pythagorean theorem tells us that:
\[ OP^2 = PA^2 + OA^2 \]

Step 2: Substituting the known values:

We are given that \( PA = 16 \, \text{cm} \) and \( OP = 20 \, \text{cm} \). Let the radius of the larger circle be \( r \). Substituting these values into the equation:
\[ 20^2 = 16^2 + r^2 \] Simplifying the equation:
\[ 400 = 256 + r^2 \] \[ r^2 = 400 - 256 = 144 \] Taking the square root of both sides:
\[ r = \sqrt{144} = 12 \, \text{cm} \] Thus, the radius of the larger circle is \( 12 \, \text{cm} \).

Step 3: Using the formula for the length of the chord:

The length of the chord \( CD \) is given by the formula:
\[ CD = 2\sqrt{OP^2 - OQ^2} \] where \( OP \) is the distance from the center to the point of tangency, and \( OQ \) is the distance from the center to the other point on the chord.
Substitute the values \( OP = 20 \, \text{cm} \) and \( OQ = 6 \, \text{cm} \):
\[ CD = 2\sqrt{20^2 - 6^2} = 2\sqrt{400 - 36} = 2\sqrt{364} \] Simplifying further:
\[ CD = 2 \times 19.08 = 38.16 \, \text{cm} \]

Step 4: Conclusion:

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