If `DeltaABC` is similar to `DeltaDEF` such that `BC = 3cm, EF=4cm` and area of `DeltaABC=54cm^(2)`. Find the area of `DeltaDEF`.

Answer

To find the area of triangle DEF given that triangle ABC is similar to triangle DEF, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between the triangles**: Since triangle ABC is similar to triangle DEF, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

2. **Identify the corresponding sides**: We know that:
- BC (side of triangle ABC) = 3 cm
- EF (corresponding side of triangle DEF) = 4 cm

3. **Calculate the ratio of the corresponding sides**:
\[ \text{Ratio of sides} = \frac{BC}{EF} = \frac{3}{4} \]

4. **Square the ratio of the sides**:
\[ \left(\frac{BC}{EF}\right)^2 = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]

5. **Set up the ratio of the areas**: Let the area of triangle DEF be \( A_{DEF} \). According to the property of similar triangles:
\[ \frac{Area_{ABC}}{Area_{DEF}} = \frac{9}{16} \]

6. **Substitute the known area of triangle ABC**: We know that the area of triangle ABC is 54 cm².
\[ \frac{54}{A_{DEF}} = \frac{9}{16} \]

7. **Cross-multiply to solve for \( A_{DEF} \)**:
\[ 54 \cdot 16 = 9 \cdot A_{DEF} \]
\[ 864 = 9 \cdot A_{DEF} \]

8. **Divide both sides by 9**:
\[ A_{DEF} = \frac{864}{9} = 96 \text{ cm}^2 \]

### Final Answer:
The area of triangle DEF is \( 96 \text{ cm}^2 \).

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