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 make the arithmetic easier and prevent decimal mistakes, you can express the decimal radius as a fraction:
\[ r = 6.3 = \frac{63}{10} \text{ cm} \] Substituting this fraction makes cancelling terms much simpler:
\[ 11 = \frac{\theta}{360^\circ} \times 2 \times \frac{22}{7} \times \frac{63}{10} \] \[ 11 = \frac{\theta}{360^\circ} \times \frac{44 \times 9}{10} = \frac{\theta}{360^\circ} \times \frac{396}{10} \] \[ 11 = \frac{36 \times 11 \times \theta}{3600} \implies 1 = \frac{\theta}{100} \implies \theta = 100^\circ \] This method is highly recommended for speed and accuracy!

Answer 1

The Correct Option is D

Step 1: Use the radian form of the arc length formula.
In radians, arc length is $l = r\theta$, so $\theta$ (in radians) $= \frac{l}{r} = \frac{11}{6.3}$.
Step 2: Convert radians to degrees.
$\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180^\circ}{\pi} = \frac{11}{6.3} \times \frac{180 \times 7}{22}$.
Step 3: Simplify the fraction.
\[ \theta = \frac{11 \times 180 \times 7}{6.3 \times 22} = \frac{13860}{138.6} \]
Dividing numerator and denominator by 138.6 gives $\theta = 100^\circ$.
\[ \boxed{\theta = 100^\circ} \]