A Circular Wire of Radius 7 cm is Cut and Bent Again into an Arc of a Circle of Radius 12 cm

Question:

A circular wire of radius \(7\) cm is cut and bent again into an arc of a circle of radius \(12\) cm. The angle subtended by the arc at the center is

(a) \(50^\circ\)

(b) \(210^\circ\)

(c) \(100^\circ\)

(d) \(60^\circ\)

(e) \(195^\circ\)

Solution

Radius of original circle:

\[ r = 7 \text{ cm} \]

Length of the wire equals the circumference of the original circle:

\[ L = 2\pi r \]

\[ L = 2\pi \times 7 \]

\[ L = 14\pi \text{ cm} \]

This wire is bent into an arc of a circle of radius \(12\) cm.

Using arc length formula:

\[ L = R\theta \]

\[ 14\pi = 12\theta \]

\[ \theta = \frac{14\pi}{12} \]

\[ = \frac{7\pi}{6} \]

Converting into degrees:

\[ \frac{7\pi}{6} \times \frac{180^\circ}{\pi} \]

\[ = 210^\circ \]

Hence, the correct option is:

(b) \(210^\circ\)