Question

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

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

Hint:

To avoid working with decimals, you can express the decimal radius as a fraction:
\[ r = 6.3 = \frac{63}{10} \text{ cm} \] Substituting this into the formula makes the calculations much easier:
\[ 11 = \frac{\theta}{360} \times 2 \times \frac{22}{7} \times \frac{63}{10} \] \[ 11 = \frac{\theta}{360} \times \frac{44 \times 9}{10} \] \[ 11 = \frac{\theta \times 396}{3600} \implies 1 = \frac{\theta \times 36}{3600} \implies 1 = \frac{\theta}{100} \implies \theta = 100^\circ \] Converting decimals to fractions prevents simple arithmetic mistakes!

Answer 1

The Correct Option is D

Step 1: Find the full circumference of the circle.
\[ \text{Circumference} = 2\pi r = 2 \times \frac{22}{7} \times 6.3 = 39.6 \text{ cm} \]
Step 2: Use the idea that arc length is a fraction of the circumference.
The fraction of the full circle that this arc represents equals the fraction that \(\theta\) is of \(360^\circ\): \[ \frac{\theta}{360^\circ} = \frac{\text{arc length}}{\text{circumference}} = \frac{11}{39.6} \]
Step 3: Simplify this fraction.
\[ \frac{11}{39.6} = \frac{110}{396} = \frac{5}{18} \]
Step 4: Solve for theta.
\[ \theta = 360^\circ \times \frac{5}{18} = 100^\circ \]
This matches option (D).
\[ \boxed{\theta = 100^\circ} \]