In a circle of diameter44 cm, the length of a chord is 22 cm. What is the length of minor arc of the chord ?

Options

A. `(484)/(21)` cm
B. `(242)/(21)`cm
C. `(121)/(21)`cm
D. `(44)/(7)`cm

Explanation

To find the length of the minor arc of a chord in a circle, we can follow these steps:

### Step 1: Find the radius of the circle
Given the diameter of the circle is 44 cm, we can find the radius using the formula:
\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{44 \text{ cm}}{2} = 22 \text{ cm} \]

### Step 2: Use the chord length to find the angle subtended at the center
The length of the chord is given as 22 cm. We can use the relationship between the chord length, radius, and the angle subtended at the center of the circle. For a chord length \( c \) and radius \( r \), the angle \( \theta \) in radians can be found using the formula:
\[ c = 2r \sin\left(\frac{\theta}{2}\right) \]
Substituting the known values:
\[ 22 = 2 \times 22 \sin\left(\frac{\theta}{2}\right) \]
This simplifies to:
\[ 22 = 44 \sin\left(\frac{\theta}{2}\right) \]
\[ \sin\left(\frac{\theta}{2}\right) = \frac{22}{44} = \frac{1}{2} \]
Thus,
\[ \frac{\theta}{2} = 30^\circ \implies \theta = 60^\circ \]

### Step 3: Convert the angle to radians
To find the length of the arc, we need the angle in radians. We convert degrees to radians using the conversion factor \( \frac{\pi}{180} \):
\[ \theta \text{ (in radians)} = 60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians} \]

### Step 4: Calculate the length of the minor arc
The length of the arc \( L \) can be calculated using the formula:
\[ L = r \theta \]
Substituting the values:
\[ L = 22 \times \frac{\pi}{3} \]
\[ L = \frac{22\pi}{3} \text{ cm} \]

### Step 5: Approximate the value (if needed)
If we want to approximate the value of the arc length using \( \pi \approx \frac{22}{7} \):
\[ L \approx \frac{22 \times \frac{22}{7}}{3} = \frac{484}{21} \text{ cm} \]

Thus, the length of the minor arc of the chord is:
\[ \frac{22\pi}{3} \text{ cm} \text{ or approximately } \frac{484}{21} \text{ cm} \]