An arc subtended an angle `60^(@)` at the center of c circle of radius 6 cm , then length of minor and major arc

Options

A. ` 2 pi and 10 pi`
B. `10 pi and 2 pi`
C. `8pi and 4pi`
D. `4 pi and 6 pi`

Answer

To find the lengths of the minor and major arcs of a circle subtended by an angle of \(60^\circ\) at the center with a radius of \(6 \, \text{cm}\), we can follow these steps:

### Step 1: Understand the formula for the length of an arc
The length of an arc \(L\) can be calculated using the formula:
\[ L = \frac{\theta}{360^\circ} \times 2\pi r \]
where:
- \(L\) is the length of the arc,
- \(\theta\) is the angle subtended at the center (in degrees),
- \(r\) is the radius of the circle.

### Step 2: Calculate the length of the minor arc
Given:
- \(\theta = 60^\circ\)
- \(r = 6 \, \text{cm}\)

Substituting the values into the formula:
\[ L_{\text{minor}} = \frac{60^\circ}{360^\circ} \times 2\pi \times 6 \]

Calculating:
\[ L_{\text{minor}} = \frac{1}{6} \times 2\pi \times 6 = \frac{12\pi}{6} = 2\pi \, \text{cm} \]

### Step 3: Calculate the length of the major arc
The major arc subtended by the angle can be calculated by subtracting the angle of the minor arc from \(360^\circ\):
\[ \theta_{\text{major}} = 360^\circ - 60^\circ = 300^\circ \]

Now, using the arc length formula again:
\[ L_{\text{major}} = \frac{300^\circ}{360^\circ} \times 2\pi \times 6 \]

Calculating:
\[ L_{\text{major}} = \frac{5}{6} \times 2\pi \times 6 = \frac{60\pi}{6} = 10\pi \, \text{cm} \]

### Final Answer
- Length of the minor arc: \(2\pi \, \text{cm}\)
- Length of the major arc: \(10\pi \, \text{cm}\)

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