Question

Arc PQ subtends an angle \(\theta\) at the centre of the circle with radius 6.3 cm. If 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:
\[ 11 = \frac{\theta}{360} \times 2 \times \frac{22}{7} \times \frac{63}{10} \] \[ 11 = \frac{\theta}{360} \times \frac{44 \times 9}{10} = \frac{\theta}{360} \times \frac{396}{10} \] \[ 11 = \frac{11 \times 36 \times \theta}{3600} \implies 1 = \frac{\theta}{100} \implies \theta = 100^\circ \] Converting decimals to fractions prevents simple arithmetic mistakes!

Answer 1

The Correct Option is D

Step 1: Rewrite the radius as a clean fraction to avoid decimals.
$r=6.3=\frac{63}{10}$ cm.
Step 2: Substitute into the arc-length formula.
$11=\frac{\theta}{360^\circ}\times2\times\frac{22}{7}\times\frac{63}{10}$.
Step 3: Simplify the constants first.
$2\times\frac{22}{7}\times\frac{63}{10}=\frac{2\times22\times9}{10}=39.6$, so $11=\frac{\theta}{360^\circ}\times39.6$.
Step 4: Solve for $\theta$.
$\theta=\frac{11\times360^\circ}{39.6}=100^\circ$, matching option (D).
\[ \boxed{100^\circ} \]