Ratio of corresponding sides of two similar triangles is 2 : 5. if the area of the smaller triangle is 64sq. Cm , then what is the area of the bigger triangle?

Answer

To solve the problem step by step, we will use the properties of similar triangles and the relationship between the ratios of their sides and areas.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- The ratio of the corresponding sides of two similar triangles is \(2 : 5\).
- The area of the smaller triangle (let's call it \(A_1\)) is \(64 \, \text{cm}^2\).

2. **Set Up the Ratio of Areas**:
- For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.
- Therefore, if the ratio of the sides is \( \frac{2}{5} \), then the ratio of the areas \( \frac{A_1}{A_2} \) (where \(A_2\) is the area of the bigger triangle) is given by:
\[ \frac{A_1}{A_2} = \left(\frac{2}{5}\right)^2 \]

3. **Calculate the Square of the Side Ratio**:
- Calculate \( \left(\frac{2}{5}\right)^2 \):
\[ \left(\frac{2}{5}\right)^2 = \frac{4}{25} \]

4. **Set Up the Equation**:
- Substitute \(A_1 = 64\) into the ratio:
\[ \frac{64}{A_2} = \frac{4}{25} \]

5. **Cross Multiply to Solve for \(A_2\)**:
- Cross multiplying gives:
\[ 64 \times 25 = 4 \times A_2 \]

6. **Calculate \(64 \times 25\)**:
- Calculate \(64 \times 25\):
\[ 64 \times 25 = 1600 \]
- So, the equation now is:
\[ 1600 = 4 \times A_2 \]

7. **Solve for \(A_2\)**:
- Divide both sides by \(4\):
\[ A_2 = \frac{1600}{4} = 400 \]

8. **Conclusion**:
- The area of the bigger triangle is \(400 \, \text{cm}^2\).

### Final Answer:
The area of the bigger triangle is \(400 \, \text{cm}^2\).