Question

Arc \(PQ\) subtends an angle \(\theta\) at the centre of the circle with radius \(6.3 \text{ cm}\). If \(\text{Arc } PQ = 11 \text{ cm}\), then the value of \(\theta\) is

• A. \(10^{\circ}\)
• B. \(60^{\circ}\)
• C. \(45^{\circ}\)
• D. \(100^{\circ}\)

Hint:

When dealing with decimals like 6.3 in arc length or area of sector problems, they are usually multiples of 7. Using \(\pi = 22/7\) often leads to easy cancellations.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The length of an arc is proportional to the angle it subtends at the center of the circle.
Step 2: Key Formula or Approach:
Length of Arc (\(l\)) = \(\frac{\theta}{360^{\circ}} \times 2 \pi r\)
Given: \(l = 11 \text{ cm}\), \(r = 6.3 \text{ cm}\), \(\pi \approx \frac{22}{7}\).
Step 3: Detailed Explanation:
Substitute the given values into the formula:
\[ 11 = \frac{\theta}{360^{\circ}} \times 2 \times \frac{22}{7} \times 6.3 \]
\[ 11 = \frac{\theta}{360^{\circ}} \times 44 \times 0.9 \]
\[ 11 = \frac{\theta}{360^{\circ}} \times 39.6 \]
Divide both sides by 11:
\[ 1 = \frac{\theta}{360^{\circ}} \times 3.6 \]
\[ 360^{\circ} = 3.6\theta \]
\[ \theta = \frac{360}{3.6} = 100^{\circ} \]
Step 4: Final Answer:
The value of \(\theta\) is \(100^{\circ}\).