Question

Chord AB subtends an angle of 120º at the centre O of the circle with radius \(\frac{21}{2}\) cm. Find the perimeter of shaded segment ACB. (Use \(\sqrt{3} = 1.7\))

Hint:

Always use the exact value of \(\sqrt{3}\) specified in the question.
Using the standard \(1.73\) instead of the given \(1.7\) will result in a slightly different decimal answer, which can lead to a loss of marks!

Answer 1

Step 1: Understanding the Question:
The topic of this question is Areas and Perimeters Related to Circles.
The shaded region in the figure is a segment of the circle.
The perimeter (boundary) of this segment is made up of two distinct parts:
1. The straight line segment, which is the chord \(AB\).
2. The curved boundary, which is the arc length of the minor arc \(AB\).
We are given the radius \(r = \frac{21}{2} = 10.5\) cm and the central angle \(\theta = 120^\circ\). We need to calculate both lengths and add them.

Step 2: Key Formula or Approach:
- Formula for the length of an arc \(l\) that subtends an angle \(\theta\) at the center:
\[ l = \frac{\theta}{360^\circ} \times 2\pi r \] - Formula for the length of a chord \(AB\) that subtends an angle \(\theta\) at the center:
\[ AB = 2r \sin\left(\frac{\theta}{2}\right) \] - The total perimeter of the shaded region is:
\[ \text{Perimeter} = \text{Arc Length } AB + \text{Chord Length } AB \]

Step 3: Detailed Explanation:

• Calculate the length of the minor arc \(AB\):
Substitute the given values \(r = \frac{21}{2}\) cm and \(\theta = 120^\circ\) into the arc length formula:
\[ l = \frac{120^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times \frac{21}{2} \] \[ l = \frac{1}{3} \times \frac{44 \times 21}{14} \] \[ l = \frac{1}{3} \times 22 \times 3 = 22 \text{ cm} \]

• Calculate the length of the chord \(AB\):
Substitute \(r = \frac{21}{2}\) cm and \(\theta = 120^\circ\) into the chord length formula:
\[ AB = 2\left(\frac{21}{2}\right) \sin\left(\frac{120^\circ}{2}\right) \] \[ AB = 21 \sin(60^\circ) \] \[ AB = 21 \times \frac{\sqrt{3}}{2} \] \[ AB = 10.5\sqrt{3} \text{ cm} \]

• Substitute the given value \(\sqrt{3} = 1.7\) to convert the chord length to a decimal value:
\[ AB = 10.5 \times 1.7 = 17.85 \text{ cm} \]

• Calculate the total perimeter of the shaded segment by adding the two values:
\[ \text{Perimeter} = \text{Arc Length} + \text{Chord Length} \] \[ \text{Perimeter} = 22 + 17.85 = 39.85 \text{ cm} \]

Step 4: Final Answer:
The perimeter of the shaded region is 39.85 cm.