Areas of two similar triangles are i the ratio of 4:5, then the ratio of their corresponding sides is :

Options

A. `4:5`
B. `16:25`
C. `2:sqrt(5)`
D. `5:4`

Answer

To solve the problem of finding the ratio of the corresponding sides of two similar triangles given that their areas are in the ratio of 4:5, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Areas and Sides of Similar Triangles**:
- For two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. If the areas of the triangles are in the ratio \( A_1 : A_2 \), then the ratio of their corresponding sides \( S_1 : S_2 \) can be expressed as:
\[ \frac{A_1}{A_2} = \left(\frac{S_1}{S_2}\right)^2 \]

2. **Set Up the Given Ratio**:
- We are given that the areas of the two triangles are in the ratio of 4:5. Hence, we can write:
\[ \frac{A_1}{A_2} = \frac{4}{5} \]

3. **Apply the Relationship**:
- According to the property of similar triangles, we can equate this to the square of the ratio of the sides:
\[ \frac{4}{5} = \left(\frac{S_1}{S_2}\right)^2 \]

4. **Take the Square Root**:
- To find the ratio of the sides, we take the square root of both sides:
\[ \frac{S_1}{S_2} = \sqrt{\frac{4}{5}} \]

5. **Simplify the Square Root**:
- Simplifying the square root gives:
\[ \frac{S_1}{S_2} = \frac{\sqrt{4}}{\sqrt{5}} = \frac{2}{\sqrt{5}} \]

6. **Final Result**:
- Therefore, the ratio of the corresponding sides of the two triangles is:
\[ \frac{S_1}{S_2} = \frac{2}{\sqrt{5}} \]