`DeltaABC~DeltaEDF` and AB = 5 cm, BC = 8 cm and AC = 10 cm. If ar `(DeltaABC)` : ar`(DeltaDEF)` = 9 : 4 , then DF is equal to:

Options

A. `16/3cm`
B. `32/9cm`
C. `10/3cm`
D. `20/3cm`

Answer

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

### Step-by-Step Solution:

1. **Understand the Similarity of Triangles**:
Since \( \Delta ABC \sim \Delta EDF \), the ratios of the areas of the triangles are equal to the square of the ratios of their corresponding sides.

2. **Write the Area Ratio**:
We are given that the ratio of the areas of triangles \( ABC \) and \( DEF \) is:
\[ \frac{ar(\Delta ABC)}{ar(\Delta DEF)} = \frac{9}{4} \]

3. **Set Up the Ratio of Sides**:
Let \( DF \) be the side corresponding to \( BC \) in triangle \( ABC \). The ratio of the sides can be expressed as:
\[ \left( \frac{BC}{DF} \right)^2 = \frac{9}{4} \]

4. **Substitute the Length of Side BC**:
We know \( BC = 8 \, \text{cm} \). Substitute this value into the equation:
\[ \left( \frac{8}{DF} \right)^2 = \frac{9}{4} \]

5. **Take the Square Root**:
Taking the square root of both sides gives:
\[ \frac{8}{DF} = \frac{3}{2} \]

6. **Cross-Multiply to Solve for DF**:
Cross-multiplying yields:
\[ 8 \cdot 2 = 3 \cdot DF \]
\[ 16 = 3 \cdot DF \]

7. **Isolate DF**:
To find \( DF \), divide both sides by 3:
\[ DF = \frac{16}{3} \, \text{cm} \]

### Final Answer:
Thus, the length of \( DF \) is \( \frac{16}{3} \, \text{cm} \).

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