Which of the following statement is incorrect?

Options

A. The ratio of perimeters of two similar `Deltas` is the same as the ratio of their corresponding sides.
B. If the areas of two similar `Deltas` are equal, then they are congurent.
C. If the ratio of areas of two similar `Deltas` is equal to the ratio of the sides.
D. If ratio of corresponding is `5 : 8`, then ratio of their areas are `25 : 64`.

Answer

To determine which statement is incorrect among the given options related to similar triangles, we will analyze each statement step by step.

### Step-by-Step Solution:

1. **Understanding Similar Triangles:**
- Similar triangles have the same shape but may differ in size. The corresponding angles are equal, and the lengths of corresponding sides are in proportion.

2. **Analyzing Option 1:**
- **Statement:** The ratio of the perimeter of two similar triangles is the same as the ratio of their corresponding sides.
- **Verification:**
- If we have two similar triangles with corresponding sides in the ratio \( a:b \), then their perimeters will also be in the ratio \( a:b \).
- Example: If the sides of the first triangle are 3, 4, and 6, and the corresponding sides of the second triangle are 6, 8, and 12, the perimeter of the first triangle is \( 3 + 4 + 6 = 13 \) and the second triangle is \( 6 + 8 + 12 = 26 \). The ratio of perimeters is \( 13:26 = 1:2 \), which matches the ratio of sides \( 3:6 = 1:2 \).
- **Conclusion:** This statement is **correct**.

3. **Analyzing Option 2:**
- **Statement:** If the areas of two similar triangles are equal, then they are congruent.
- **Verification:**
- For two triangles to be congruent, they must have the same shape and size. If two similar triangles have equal areas, it implies that they are indeed congruent, as congruence means that all corresponding sides and angles are equal.
- **Conclusion:** This statement is **correct**.

4. **Analyzing Option 3:**
- **Statement:** The ratio of the areas of two similar triangles is equal to the ratio of their sides.
- **Verification:**
- The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. If the ratio of the sides is \( a:b \), then the ratio of the areas is \( a^2:b^2 \).
- Example: If the sides of the first triangle are 3 and 6, the ratio of sides is \( 1:2 \). The ratio of areas would be \( 1^2:2^2 = 1:4 \), not \( 1:2 \).
- **Conclusion:** This statement is **incorrect**.

5. **Analyzing Option 4:**
- **Statement:** (Assuming a statement exists for analysis)
- **Verification:** (Would need the actual statement to analyze)
- **Conclusion:** (Would need the actual statement to conclude)

### Final Conclusion:
The incorrect statement among the options is **Option 3**: "The ratio of the areas of two similar triangles is equal to the ratio of their sides."