In `DeltaABC`, D and E are the points of sides AB and BC respectively such that DE || AC and AD : DB = 3:2. The ratio of area of trapezium `Delta CEDA` to that of `Delta BED` is 

Options

A. `4:15`
B. `15:4`
C. `4:21`
D. `21:4`

Answer

To solve the problem, we need to find the ratio of the area of trapezium CEDA to that of triangle BED in triangle ABC, given that DE is parallel to AC and AD:DB = 3:2.

### Step-by-Step Solution:

1. **Understanding the Ratios**:
- We are given that AD:DB = 3:2. This means that if we let AD = 3x and DB = 2x, then AB = AD + DB = 3x + 2x = 5x.

2. **Using the Properties of Similar Triangles**:
- Since DE is parallel to AC, triangles CED and CAB are similar by the Basic Proportionality Theorem (also known as Thales' theorem). This implies that the ratios of their corresponding sides are equal.

3. **Finding 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.
- The ratio of the sides corresponding to triangles CED and CAB is AD:AB = 3x:5x = 3:5.
- Therefore, the ratio of the areas of triangles CED and CAB is (3/5)² = 9/25.

4. **Finding the Area of Triangle BED**:
- The area of triangle BED can be found by considering the area of triangle ABC and subtracting the area of triangle CED from it.
- Let the area of triangle ABC be A. Then, the area of triangle CED is (9/25)A.
- The area of triangle BED will be A - (9/25)A = (25/25)A - (9/25)A = (16/25)A.

5. **Finding the Area of Trapezium CEDA**:
- The area of trapezium CEDA is the area of triangle ABC minus the area of triangle BED.
- Thus, Area of trapezium CEDA = A - (16/25)A = (25/25)A - (16/25)A = (9/25)A.

6. **Finding the Ratio of Areas**:
- Now we have the areas:
- Area of trapezium CEDA = (9/25)A
- Area of triangle BED = (16/25)A
- Therefore, the ratio of the area of trapezium CEDA to the area of triangle BED is:
\[ \text{Ratio} = \frac{\text{Area of CEDA}}{\text{Area of BED}} = \frac{(9/25)A}{(16/25)A} = \frac{9}{16}. \]

### Final Answer:
The ratio of the area of trapezium CEDA to that of triangle BED is **9:16**.