Question

An arc of length \(2.2\) cm subtends an angle \(\theta\) at the centre of the circle with radius \(2.8\) cm. The value of \(\theta\) is

• A. \(50^{\circ}\)
• B. \(60^{\circ}\)
• C. \(45^{\circ}\)
• D. \(30^{\circ}\)

Hint:

Notice that \(17.6\) is exactly \(8\) times \(2.2\). Identifying such ratios quickly makes simplifying fractions much easier in competitive exams.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The length of an arc of a circle is proportional to the angle it subtends at the center. The total circumference subtends \(360^{\circ}\).
Step 2: Key Formula or Approach:
Length of arc (\(l\)) is given by:
\[ l = \frac{\theta}{360^{\circ}} \times 2\pi r \]
Step 3: Detailed Explanation:
Given:
Arc length \(l = 2.2\) cm
Radius \(r = 2.8\) cm
Using \(\pi = \frac{22}{7}\):
\[ 2.2 = \frac{\theta}{360^{\circ}} \times 2 \times \frac{22}{7} \times 2.8 \]
\[ 2.2 = \frac{\theta}{360^{\circ}} \times 44 \times 0.4 \]
\[ 2.2 = \frac{\theta}{360^{\circ}} \times 17.6 \]
Rearranging to solve for \(\theta\):
\[ \theta = \frac{2.2 \times 360^{\circ}}{17.6} \]
\[ \theta = \frac{360^{\circ}}{8} = 45^{\circ} \]
Step 4: Final Answer:
The value of \(\theta\) is \(45^{\circ}\).