D and E are points on sides AB and AC of `Delta ABC`. DE is parallel to BC. If AD : DB = 2 : 3, what is the ratio of area `Delta ADE` and area of quadrilateral BDEC ?

Options

A. `4:21`
B. `4:25`
C. `4:29`
D. `4:9`

Answer

To solve the problem, we need to find the ratio of the area of triangle ADE to the area of quadrilateral BDEC, given that DE is parallel to BC and the ratio AD:DB is 2:3.

### Step-by-Step Solution:

1. **Understanding the Triangle and Points**:
- Let triangle ABC be given with points D on AB and E on AC such that DE is parallel to BC.
- Since DE is parallel to BC, triangles ADE and ABC are similar by the Basic Proportionality Theorem (also known as Thales' theorem).

2. **Setting Up Ratios**:
- We know that AD:DB = 2:3. This means that if we let AD = 2x and DB = 3x, then AB = AD + DB = 2x + 3x = 5x.

3. **Finding the Ratio of Sides**:
- Since DE is parallel to BC, the ratio of the corresponding sides of the similar triangles ADE and ABC is the same as the ratio of AD to AB.
- Therefore, the ratio of the sides is:
\[ \frac{AD}{AB} = \frac{2x}{5x} = \frac{2}{5} \]

4. **Calculating the Area Ratio**:
- The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore:
\[ \frac{\text{Area of } \triangle ADE}{\text{Area of } \triangle ABC} = \left(\frac{AD}{AB}\right)^2 = \left(\frac{2}{5}\right)^2 = \frac{4}{25} \]

5. **Finding the Area of Quadrilateral BDEC**:
- The area of quadrilateral BDEC can be found by subtracting the area of triangle ADE from the area of triangle ABC:
\[ \text{Area of } BDEC = \text{Area of } \triangle ABC - \text{Area of } \triangle ADE \]
- Let the area of triangle ABC be represented as 25k (for some constant k), then the area of triangle ADE is:
\[ \text{Area of } \triangle ADE = \frac{4}{25} \times 25k = 4k \]
- Thus, the area of quadrilateral BDEC is:
\[ \text{Area of } BDEC = 25k - 4k = 21k \]

6. **Finding the Final Ratio**:
- Now we can find the ratio of the area of triangle ADE to the area of quadrilateral BDEC:
\[ \frac{\text{Area of } \triangle ADE}{\text{Area of } BDEC} = \frac{4k}{21k} = \frac{4}{21} \]

### Final Answer:
The ratio of the area of triangle ADE to the area of quadrilateral BDEC is \( \frac{4}{21} \).