Question

Chord AB of a circle subtends an angle of \(120^\circ\) at the centre O of the circle. Find the length of arc AB, if radius of the circle is 21 cm.

Hint:

Notice that \(\frac{120^\circ}{360^\circ}\) is exactly \(\frac{1}{3}\) of the circle.
The total circumference of the circle is \(2\pi r = 2 \times \frac{22}{7} \times 21 = 132\text{ cm}\).
Dividing 132 by 3 gives 44 cm, which is a fast way to verify your result!

Answer 1

Correct Answer: 44

Step 1: Understanding the Question:
This question is from the chapter Areas Related to Circles.
We are given a circle with a radius of \(21\text{ cm}\) and a chord \(AB\) that subtends an angle of \(120^\circ\) at the centre \(O\).
We need to calculate the length of the corresponding arc \(AB\).

Step 2: Key Formula or Approach:
The length of an arc (\(l\)) subtending an angle \(\theta\) at the centre of a circle of radius \(r\) is given by the formula: \[ l = \frac{\theta}{360^\circ} \times 2\pi r \] where: - \(\theta = 120^\circ\)
- \(r = 21\text{ cm}\)
- \(\pi = \frac{22}{7}\)

Step 3: Detailed Explanation:
1. Write down the given values from the problem description: - Radius of the circle, \(r = 21\text{ cm}\)
- Angle subtended at the centre, \(\theta = 120^\circ\)
2. Substitute these values into the arc length formula: \[ l = \frac{120^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 21 \] 3. Simplify the terms: - Simplify the angle fraction: \[ \frac{120^\circ}{360^\circ} = \frac{1}{3} \] - Substitute this back into the equation: \[ l = \frac{1}{3} \times 2 \times \frac{22}{7} \times 21 \] 4. Simplify the numerical terms: - Since \(3 \times 7 = 21\), the denominator terms cancel out the term 21 in the numerator: \[ l = 2 \times 22 \] \[ l = 44\text{ cm} \]

Step 4: Final Answer:
The length of the arc \(AB\) is 44 cm.