Consider the following statements Area of segment of a circle is less than area of its corresponding sector. Distance travelled by a circular Wheel of diameter 2d cm in one revolution is greater than 6d cm. Which of the above statements is/are correct?

Options

A. a. Only I
B. b. Only II
C. c. Both I and II
D. d. neither I nor II

Answer

To solve the problem, we need to analyze the two statements given in the question one by one.

### Step 1: Analyze the first statement
**Statement 1:** The area of the segment of a circle is less than the area of its corresponding sector.

- **Understanding the terms:**
- A **sector** is a portion of a circle defined by two radii and the arc between them.
- A **segment** is the area of a circle that is "cut off" from the rest of the circle by a chord.

- **Formula for the area of a sector:**
\[ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 \]
where \(\theta\) is the angle in degrees and \(r\) is the radius of the circle.

- **Formula for the area of a triangle formed by the two radii and the chord:**
\[ \text{Area of triangle} = \frac{1}{2} r^2 \sin(\theta) \]

- **Formula for the area of the segment:**
\[ \text{Area of segment} = \text{Area of sector} - \text{Area of triangle} \]

- **Conclusion:**
Since the area of the segment is the area of the sector minus the area of the triangle, it follows that:
\[ \text{Area of segment} < \text{Area of sector} \]
Therefore, **Statement 1 is correct.**

### Step 2: Analyze the second statement
**Statement 2:** The distance travelled by a circular wheel of diameter \(2d\) cm in one revolution is greater than \(6d\) cm.

- **Understanding the terms:**
- The distance travelled in one revolution of a circular wheel is equal to its circumference.

- **Formula for the circumference of a circle:**
\[ \text{Circumference} = \pi \times \text{diameter} = \pi \times 2d = 2\pi d \]

- **Calculating the circumference:**
Using the approximate value of \(\pi \approx 3.14\):
\[ \text{Circumference} \approx 2 \times 3.14 \times d = 6.28d \]

- **Conclusion:**
Since \(6.28d\) is greater than \(6d\), we can conclude that:
\[ \text{Circumference} > 6d \]
Therefore, **Statement 2 is also correct.**

### Final Conclusion
Both statements are correct.