Let `DeltaABC-DeltaDEF` and their areas be , respectively , `64cm^(2)and 121cm^(2)`. If EF=15.4 cm , find BC.

Answer

To solve the problem, we need to find the length of side BC in triangle ABC, given that triangles ABC and DEF are similar and their areas are 64 cm² and 121 cm² respectively. We also know that EF = 15.4 cm.

### Step-by-Step Solution:

1. **Identify the Areas of the Triangles:**
- Area of triangle ABC = 64 cm²
- Area of triangle DEF = 121 cm²

2. **Use the Ratio of Areas to Find the Ratio of Sides:**
Since the triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Therefore, we can write:
\[ \frac{\text{Area of } ABC}{\text{Area of } DEF} = \left(\frac{BC}{EF}\right)^2 \]
Substituting the areas:
\[ \frac{64}{121} = \left(\frac{BC}{15.4}\right)^2 \]

3. **Set Up the Equation:**
Let \( BC = x \). Then we have:
\[ \frac{64}{121} = \left(\frac{x}{15.4}\right)^2 \]

4. **Cross-Multiply to Eliminate the Fraction:**
\[ 64 \cdot (15.4)^2 = 121 \cdot x^2 \]

5. **Calculate \( (15.4)^2 \):**
\[ (15.4)^2 = 237.16 \]
Now substituting this value back into the equation:
\[ 64 \cdot 237.16 = 121 \cdot x^2 \]

6. **Calculate \( 64 \cdot 237.16 \):**
\[ 64 \cdot 237.16 = 15178.24 \]
So now we have:
\[ 15178.24 = 121 \cdot x^2 \]

7. **Solve for \( x^2 \):**
\[ x^2 = \frac{15178.24}{121} \]

8. **Calculate \( x^2 \):**
\[ x^2 = 125.24 \]

9. **Take the Square Root to Find \( x \):**
\[ x = \sqrt{125.24} \approx 11.2 \text{ cm} \]

### Final Answer:
The length of side BC is approximately **11.2 cm**.