Question

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

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

Hint:

When dealing with decimals like 6.3 in arc length or area of sector problems, they are usually multiples of 7. Using \(\pi = 22/7\) often leads to easy cancellations.

Answer 1

The Correct Option is D

To solve the problem of finding the angle \(\theta\) subtended by arc \(PQ\) at the center of the circle, we need to use the formula for arc length:

\(L = r \theta\)

Here, \(L\) is the arc length, \(r\) is the radius of the circle, and \(\theta\) is the angle in radians.

Given:

• Arc \(PQ = 11 \text{ cm}\)
• Radius \(r = 6.3 \text{ cm}\)

Using the formula, we can find \(\theta\) as follows:

\(\theta = \frac{L}{r} = \frac{11 \text{ cm}}{6.3 \text{ cm}}\)

Calculating this gives:

\(\theta \approx 1.746 \text{ radians}\)

To convert radians to degrees, we use the conversion factor:

\(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\)

Therefore,

\(\theta \approx 1.746 \times \frac{180}{\pi} \approx 100^{\circ}\)

This matches with the given correct answer option.