Question

Chord AB of a circle with centre O and radius 21 mm subtends an angle of \(120^\circ\) at the centre. Find the perimeters of the shaded region. (Use \(\sqrt{3} = 1.73\))

Hint:

You can also find the chord length in an isosceles triangle \(\Delta OAB\) with angle \(120^\circ\) by dropping a perpendicular from the center \(O\) to the chord \(AB\).
This perpendicular bisects the angle into two \(60^\circ\) angles and bisects the chord into two equal segments.
Using simple right-triangle trigonometry, each half of the chord is \(r \sin 60^\circ\), which gives the total chord length as \(2r \sin 60^\circ = r\sqrt{3}\) directly!

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 minor 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 \(r = 21\) mm 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 \(r = 21\) mm and \(\theta = 120^\circ\) into the arc length formula:
\[ l = \frac{120^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 21 \] \[ l = \frac{1}{3} \times 2 \times 22 \times 3 \] Cancel the factors of 3:
\[ l = 2 \times 22 = 44 \text{ mm} \]

• Calculate the length of the chord \(AB\):
Substitute \(r = 21\) mm and \(\theta = 120^\circ\) into the chord length formula:
\[ AB = 2r \sin\left(\frac{120^\circ}{2}\right) \] \[ AB = 2(21) \sin(60^\circ) \] \[ AB = 42 \times \frac{\sqrt{3}}{2} \] \[ AB = 21\sqrt{3} \text{ mm} \]

• Substitute the given value \(\sqrt{3} = 1.73\) to convert the chord length to a decimal value:
\[ AB = 21 \times 1.73 = 36.33 \text{ mm} \]

• Calculate the total perimeter of the shaded segment by adding the two values:
\[ \text{Perimeter} = \text{Arc Length} + \text{Chord Length} \] \[ \text{Perimeter} = 44 + 36.33 = 80.33 \text{ mm} \]

Step 4: Final Answer:
The perimeter of the shaded region is 80.33 mm.