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: Understanding the Question:
The topic of this question is Areas Related to Circles, specifically focusing on the length of an arc of a circle.
An arc is a portion of the circumference of a circle.
The length of an arc is directly proportional to the angle it subtends at the center of the circle.
We are given the radius of the circle as \(r = 6.3\) cm and the length of the arc \(PQ\) as \(11\) cm.
We need to determine the measure of the central angle \(\theta\).

Step 2: Key Formula or Approach:
The formula for the length \(l\) of an arc that subtends an angle \(\theta\) at the center of a circle of radius \(r\) is:
\[ l = \frac{\theta}{360^\circ} \times 2\pi r \] We are given the values \(l = 11\) cm and \(r = 6.3\) cm. We will substitute these values into the formula and solve for the unknown angle \(\theta\) using \(\pi = \frac{22}{7}\).

Step 3: Detailed Explanation:

• Substitute the known values into the arc length formula:
\[ 11 = \frac{\theta}{360^\circ} \times 2 \times \frac{22}{7} \times 6.3 \]

• Simplify the calculation on the right-hand side of the equation:
First, divide \(6.3\) by \(7\):
\[ \frac{6.3}{7} = 0.9 \] Now, substitute this value back into the equation:
\[ 11 = \frac{\theta}{360^\circ} \times 2 \times 22 \times 0.9 \] \[ 11 = \frac{\theta}{360^\circ} \times 44 \times 0.9 \] \[ 11 = \frac{\theta}{360^\circ} \times 39.6 \]

• Isolate the variable \(\theta\) to solve the equation:
\[ \theta = \frac{11 \times 360^\circ}{39.6} \]

• Simplify the fraction by multiplying both the numerator and the denominator by 10 to clear the decimal:
\[ \theta = \frac{11 \times 3600^\circ}{396} \]

• Divide \(396\) by \(11\):
\[ \frac{396}{11} = 36 \] This simplifies our equation to:
\[ \theta = \frac{3600^\circ}{36} \] \[ \theta = 100^\circ \]

Step 4: Final Answer:
The value of the central angle \(\theta\) is \(100^\circ\).
Therefore, the correct option is (D).