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: Apply the Pythagorean theorem to determine the radius of the larger circle:

Given that \( PA \) is tangent to the larger circle and \( OP \) represents the distance from the center \( O \) to the point of tangency \( P \). Since \( PA \) is tangent at \( A \), a right triangle \( OPA \) is formed. The sides of this triangle are:
- \( OP \), the distance from the center to the tangent point (to be calculated),
- \( PA \), the length of the tangent,
- \( OA \), the radius of the smaller circle.
The Pythagorean theorem states:
\[OP^2 = PA^2 + OA^2\]

Step 2: Substitute known values and solve for the radius:

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

Step 3: Calculate the chord length using the relevant formula:

The length of chord \( CD \) is determined 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 \( OQ = 6 \, \text{cm} \):
\[CD = 2\sqrt{20^2 - 6^2} = 2\sqrt{400 - 36} = 2\sqrt{364}\]Further simplification leads to:
\[CD = 2 \times 19.08 = 38.16 \, \text{cm}\]

Step 4: Final Result:

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