`DeltaABCandDeltaCDE` are two equilateral triangles such that D is the mid -point of BC . The ratio of the areas of `DeltaCDEandDeltaABC` is 1 : k then k =

Answer

To solve the problem, we need to find the value of \( k \) in the ratio of the areas of two equilateral triangles \( \Delta ABC \) and \( \Delta CDE \), given that \( D \) is the midpoint of \( BC \).

### Step-by-Step Solution:

1. **Understanding the Triangles**:
- We have two equilateral triangles, \( \Delta ABC \) and \( \Delta CDE \).
- Point \( D \) is the midpoint of side \( BC \) of triangle \( ABC \).

2. **Identifying Corresponding Sides**:
- Since \( D \) is the midpoint of \( BC \), we can denote the length of side \( BC \) as \( a \). Therefore, \( BD = CD = \frac{a}{2} \).

3. **Using the Property of Similar Triangles**:
- Since both triangles are equilateral, they are similar. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

4. **Setting Up the Ratio of Areas**:
- The area of triangle \( CDE \) can be expressed in terms of the side \( CD \), which is \( \frac{a}{2} \).
- The area of triangle \( ABC \) is based on the side \( a \).

5. **Calculating the Areas**:
- The area of triangle \( ABC \) is given by the formula:
\[ \text{Area}_{ABC} = \frac{\sqrt{3}}{4} a^2 \]
- The area of triangle \( CDE \) is given by:
\[ \text{Area}_{CDE} = \frac{\sqrt{3}}{4} \left(\frac{a}{2}\right)^2 = \frac{\sqrt{3}}{4} \cdot \frac{a^2}{4} = \frac{\sqrt{3}}{16} a^2 \]

6. **Finding the Ratio of Areas**:
- Now, we can find the ratio of the areas:
\[ \frac{\text{Area}_{CDE}}{\text{Area}_{ABC}} = \frac{\frac{\sqrt{3}}{16} a^2}{\frac{\sqrt{3}}{4} a^2} = \frac{\frac{1}{16}}{\frac{1}{4}} = \frac{1}{4} \]

7. **Expressing in Terms of k**:
- We are given that the ratio of the areas of \( \Delta CDE \) to \( \Delta ABC \) is \( 1 : k \).
- From our calculation, we found that:
\[ \frac{\text{Area}_{CDE}}{\text{Area}_{ABC}} = \frac{1}{4} \]
- Therefore, we can write:
\[ 1 : k = 1 : 4 \]
- This implies that \( k = 4 \).

### Final Answer:
Thus, the value of \( k \) is \( 4 \).