Three students Varun, Katik and Sid wrote a statement on a blackboard.
Varun wrote, Two rectangles are congruent, if they have same perimeter.
Kartik wrote, Two squares are congruent, if they have same area.
Sid wrote, Two circles are congruent, if they have same diameter.
Who wrote the correct statement?

Options

A. Varun and Kartik
B. Kartik and Sid
C. Varun and Sid
D. Varun, Kartik and Sid

Answer

To determine who wrote the correct statements, let's analyze each statement one by one.

### Step 1: Analyze Varun's Statement
Varun wrote: "Two rectangles are congruent if they have the same perimeter."

- **Congruent Definition**: Two shapes are congruent if they are identical in shape and size, meaning all corresponding sides and angles are equal.
- **Perimeter**: The perimeter of a rectangle is calculated as \( P = 2 \times (length + width) \).
- **Counterexample**: Consider two rectangles:
- Rectangle 1: Length = 10, Width = 5 (Perimeter = 30)
- Rectangle 2: Length = 15, Width = 0 (Perimeter = 30)

These two rectangles have the same perimeter but are not congruent because their dimensions are different.

**Conclusion**: Varun's statement is incorrect.

### Step 2: Analyze Kartik's Statement
Kartik wrote: "Two squares are congruent if they have the same area."

- **Area of a Square**: The area of a square is calculated as \( A = side^2 \).
- **Congruence in Squares**: If two squares have the same area, then their side lengths must be equal.

Let’s say:
- Square A has side length \( a \) and area \( a^2 \).
- Square B has side length \( b \) and area \( b^2 \).

If \( a^2 = b^2 \), then \( a = b \) (since lengths cannot be negative).

**Conclusion**: Kartik's statement is correct.

### Step 3: Analyze Sid's Statement
Sid wrote: "Two circles are congruent if they have the same diameter."

- **Diameter and Congruence**: The diameter of a circle is the distance across the circle through its center.
- **Area of a Circle**: The area of a circle is calculated as \( A = \pi \times (radius)^2 \).

If two circles have the same diameter, they also have the same radius. Thus, they will have the same area and be identical in shape and size.

**Conclusion**: Sid's statement is correct.

### Final Conclusion
- Varun's statement is incorrect.
- Kartik's statement is correct.
- Sid's statement is correct.

Therefore, the correct answer is that **Kartik and Sid wrote the correct statements**.