`DeltaABC and DeltaDEF` are similar such that `(AB)/(DE)=(BC)/(EF)`. Area of the two triangles are respectively `16cm^(2) and 49cm^(2)`. If `BC=2sqrt(2)cm`, then what is length of EF ?

Options

A. `3.5cm`
B. `(3.5)sqrt(2)cm`
C. `(3.5)sqrt(3)cm`
D. `7.0cm`

Answer

To solve the problem, we will use the properties of similar triangles and the relationship between the areas of similar triangles.

### Step-by-Step Solution:

1. **Understanding the Ratio of Areas**:
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. If the area of triangle ABC is \( A_1 = 16 \, cm^2 \) and the area of triangle DEF is \( A_2 = 49 \, cm^2 \), we can set up the following ratio:
\[ \frac{A_1}{A_2} = \left(\frac{AB}{DE}\right)^2 \]
Substituting the areas:
\[ \frac{16}{49} = \left(\frac{AB}{DE}\right)^2 \]

**Hint**: Remember that the ratio of the areas of similar triangles is the square of the ratio of their corresponding sides.

2. **Finding the Ratio of Corresponding Sides**:
To find the ratio of the sides, we take the square root of the area ratio:
\[ \frac{AB}{DE} = \sqrt{\frac{16}{49}} = \frac{4}{7} \]

**Hint**: When finding the ratio of sides from the ratio of areas, take the square root.

3. **Using the Side Lengths**:
We know that \( \frac{BC}{EF} = \frac{AB}{DE} \). Let’s denote \( BC = 2\sqrt{2} \, cm \) and we need to find \( EF \):
\[ \frac{BC}{EF} = \frac{4}{7} \]
Substituting \( BC \):
\[ \frac{2\sqrt{2}}{EF} = \frac{4}{7} \]

**Hint**: Set up the proportion using the known side length.

4. **Cross-Multiplying to Solve for EF**:
Cross-multiply to solve for \( EF \):
\[ 4 \cdot EF = 7 \cdot 2\sqrt{2} \]
\[ 4 \cdot EF = 14\sqrt{2} \]

**Hint**: Cross-multiplication is a useful technique to eliminate fractions.

5. **Isolating EF**:
Now, divide both sides by 4 to find \( EF \):
\[ EF = \frac{14\sqrt{2}}{4} = \frac{7\sqrt{2}}{2} \]

**Hint**: Simplifying fractions can help you find the final answer more easily.

6. **Final Answer**:
Thus, the length of \( EF \) is:
\[ EF = \frac{7\sqrt{2}}{2} \, cm \]