It is given that `DeltaABC ~ DeltaEDF` and Area `DeltaABC` : Area `DeltaDEF = 64: 25`. If AB = 16 cm, BC = 18 cm, CA= 20 cm. What is the value of EF (in cm)?

Options

A. 15
B. 12.5
C. 11.25
D. 10

Answer

To solve the problem, we need to find the length of side EF in triangle EDF, given that triangle ABC is similar to triangle EDF and the ratio of their areas is 64:25.

### Step-by-Step Solution:

1. **Understanding the Ratio of Areas**:
Since the triangles ABC and EDF are similar, the ratio of their areas is given by:
\[ \frac{\text{Area of } \Delta ABC}{\text{Area of } \Delta EDF} = \frac{64}{25} \]

2. **Using the Property of Similar Triangles**:
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore, we can write:
\[ \frac{AB^2}{ED^2} = \frac{64}{25} \]
or
\[ \frac{AC^2}{EF^2} = \frac{64}{25} \]
or
\[ \frac{BC^2}{DF^2} = \frac{64}{25} \]

3. **Finding the Length of EF**:
We will use the side AC to find EF. We know that:
\[ AC = 20 \text{ cm} \]
Therefore, we can set up the equation:
\[ \frac{20^2}{EF^2} = \frac{64}{25} \]

4. **Calculating the Square of AC**:
Calculate \(20^2\):
\[ 20^2 = 400 \]
So the equation becomes:
\[ \frac{400}{EF^2} = \frac{64}{25} \]

5. **Cross-Multiplying**:
Cross-multiply to solve for EF:
\[ 400 \cdot 25 = 64 \cdot EF^2 \]
\[ 10000 = 64 \cdot EF^2 \]

6. **Dividing Both Sides by 64**:
\[ EF^2 = \frac{10000}{64} \]
\[ EF^2 = 156.25 \]

7. **Taking the Square Root**:
Now, take the square root to find EF:
\[ EF = \sqrt{156.25} = 12.5 \text{ cm} \]

### Final Answer:
The value of EF is **12.5 cm**.