Question

Two concentric circles are of radii \(6\) cm and \(10\) cm. The length of a chord of the larger circle which touches the smaller circle is

• A. \(8\) cm
• B. \(12\) cm
• C. \(16\) cm
• D. \(18\) cm

Hint:

If a chord of a larger circle touches a smaller concentric circle, then the distance of the chord from the common center is equal to the radius of the smaller circle.

Answer 1

The Correct Option is C

Step 1: Understand the geometry of the figure.}
There are two concentric circles, which means both circles have the same center. The radius of the smaller circle is \(6\) cm and the radius of the larger circle is \(10\) cm. A chord of the larger circle touches the smaller circle, so this chord is tangent to the smaller circle.
This means that the perpendicular distance from the common center to the chord is equal to the radius of the smaller circle, that is: \[ d = 6 \text{ cm} \]
Step 2: Use the chord-length formula.}
For a circle of radius \(R\), if the perpendicular distance of a chord from the center is \(d\), then the length of the chord is: \[ 2\sqrt{R^2 - d^2} \] Here, \[ R = 10, \quad d = 6 \] So, the chord length is: \[ 2\sqrt{10^2 - 6^2} \]
Step 3: Simplify the expression.}
\[ 2\sqrt{100 - 36} \] \[ 2\sqrt{64} \] \[ 2 \times 8 = 16 \] Thus, the required length of the chord is: \[ 16 \text{ cm} \]
Step 4: Compare with the given options.}

• (A) \(8\) cm: Incorrect.
• (B) \(12\) cm: Incorrect.
• (C) \(16\) cm: Correct.
• (D) \(18\) cm: Incorrect.

Therefore, the correct answer is \((C)\) \(16\) cm.
Final Answer:} \(16\) cm.