Triangles ABC and DEF are similar. If area of `DeltaABC = 16 cm^2`, area of `DeltaDEF = 25 cm^2` and `BC = 2.3 cm,` then EF is :

Options

A. 2.875 cm
B. 2.758 cm
C. 2.578 cm
D. None of these

Answer

To find the length of side EF in triangle DEF, we can use the properties of similar triangles and the relationship between their areas and corresponding side lengths.

### Step-by-Step Solution:

1. **Understanding Similar Triangles**:
Since triangles ABC and DEF are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

2. **Setting Up the Ratio of Areas**:
Given:
- Area of triangle ABC = 16 cm²
- Area of triangle DEF = 25 cm²

The ratio of the areas can be expressed as:
\[ \frac{\text{Area of } ABC}{\text{Area of } DEF} = \frac{16}{25} \]

3. **Finding the Ratio of Corresponding Sides**:
Let the ratio of the corresponding sides (AB to DE, BC to EF, etc.) be \( k \). Then:
\[ \frac{AB}{DE} = \frac{BC}{EF} = k \]
Since the areas are in the ratio of \( 16:25 \), we can write:
\[ k^2 = \frac{16}{25} \]
Taking the square root of both sides gives:
\[ k = \frac{4}{5} \]

4. **Using the Ratio to Find EF**:
We know that \( BC = 2.3 \) cm. Using the ratio \( k \):
\[ \frac{BC}{EF} = \frac{4}{5} \]
Substituting \( BC = 2.3 \) cm into the equation:
\[ \frac{2.3}{EF} = \frac{4}{5} \]

5. **Cross Multiplying to Solve for EF**:
Cross-multiplying gives:
\[ 2.3 \cdot 5 = 4 \cdot EF \]
Simplifying this:
\[ 11.5 = 4 \cdot EF \]

6. **Isolating EF**:
Now, divide both sides by 4 to find EF:
\[ EF = \frac{11.5}{4} = 2.875 \text{ cm} \]

### Final Answer:
Thus, the length of side EF is \( 2.875 \) cm.

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