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

To find the angle \(\theta\) subtended by an arc at the center of a circle, we use the formula for arc length:

\(l = r\theta\)

where:

• \(l\) is the arc length,
• \(r\) is the radius of the circle,
• \(\theta\) is the angle in radians.

Given:

• Arc length, \(l = 2.2 \, \text{cm}\)
• Radius, \(r = 2.8 \, \text{cm}\)

We need to find \(\theta\) in radians first:

\(\theta = \frac{l}{r} = \frac{2.2}{2.8}\)

Calculating the above gives:

\(\theta = \frac{2.2}{2.8} = \frac{11}{14} \approx 0.7857 \, \text{radians}\)

Now, convert \(\theta\) from radians to degrees. Use the conversion formula:

\(\theta \, (\text{degrees}) = \theta \, (\text{radians}) \times \frac{180}{\pi}\)

Substitute the values:

\(\theta \, (\text{degrees}) = 0.7857 \times \frac{180}{\pi}\)

Using the approximation \(\pi \approx 3.14\), calculate:

\(\theta \, (\text{degrees}) \approx 0.7857 \times \frac{180}{3.14} \approx 45^{\circ}\)

Therefore, the angle \(\theta\) is approximately \(45^{\circ}\).

Hence, the correct answer is \(45^{\circ}\).