Question

OAB is sector of a circle with centre O and radius 7 cm. If length of arc \( \widehat{AB} = \frac{22}{3} \) cm, then \( \angle AOB \) is equal to

• A. \( \left(\frac{120}{7}\right)^\circ \)
• B. \( 45^\circ \)
• C. \( 60^\circ \)
• D. \( 30^\circ \)

Answer 1

The Correct Option is C

Given:
- Sector \(OAB\) of a circle with centre \(O\) and radius \(r = 7\, \text{cm}\).
- Length of arc \(\widehat{AB} = \frac{22}{3}\, \text{cm}\).

Step 1: Arc length formula
\[\text{Arc length} = \frac{\theta}{360^\circ} \times 2 \pi r\] where \(\theta = \angle AOB\) (in degrees).

Step 2: Substitute values
\[\frac{22}{3} = \frac{\theta}{360} \times 2 \times \frac{22}{7} \times 7\] Simplify:
\[\frac{22}{3} = \frac{\theta}{360} \times 2 \times 22 = \frac{\theta}{360} \times 44\]

Step 3: Solve for \(\theta\)
\[\frac{22}{3} = \frac{44 \theta}{360}\] Multiply both sides by 360:
\[360 \times \frac{22}{3} = 44 \theta\] \[120 \times 22 = 44 \theta\] \[2640 = 44 \theta\] \[\theta = \frac{2640}{44} = 60^\circ\]

Final Answer:
\[\boxed{60^\circ}\]