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: Understanding the Question:
The topic of this question is Areas Related to Circles, specifically focusing on the length of an arc of a circle.
The boundary of any circle has a length equal to its circumference, which is \(2\pi r\).
An arc is a portion of this circumference. The length of an arc is proportional to the central angle \(\theta\) it subtends at the center.
We are given a circle of radius \(r = 6.3\) cm, and an arc \(PQ\) of length \(11\) cm.
We need to determine the measure of the central angle \(\theta\).

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

Step 3: Detailed Explanation:

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

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

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

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

• Simplify the division step-by-step:
We know that \(396\) is divisible by \(11\):
\[ \frac{396}{11} = 36 \] Substitute this back:
\[ \theta = \frac{3600^\circ}{36} = 100^\circ \]

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