In a `DeltaABC`, D and E are two points on AB and AC respectively, and DE ∥ BC. If the length of AD and AB is 5 cm and 10 cm respectively, then find the ratio between the area of `Delta`ABC and area of Quadrilateral DECB.

Options

A. `4:1`
B. `3:5`
C. `4:3`
D. `5:2`

Answer

To solve the problem, we need to find the ratio between the area of triangle ABC and the area of quadrilateral DECB, given that DE is parallel to BC and the lengths of AD and AB are 5 cm and 10 cm respectively.

### Step-by-Step Solution:

1. **Identify the given values**:
- Length of AD = 5 cm
- Length of AB = 10 cm

2. **Calculate the ratio of AD to AB**:
\[ \frac{AD}{AB} = \frac{5}{10} = \frac{1}{2} \]

3. **Use the properties of similar triangles**:
Since DE is parallel to BC, triangles ADE and ABC are similar. The ratio of their corresponding sides will be the same:
\[ \frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC} = \frac{1}{2} \]

4. **Find the ratio of the areas of triangles ADE and ABC**:
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides:
\[ \text{Area ratio} = \left(\frac{AD}{AB}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
Therefore, the area of triangle ADE is \(x\) and the area of triangle ABC is \(4x\).

5. **Calculate the area of quadrilateral DECB**:
The area of quadrilateral DECB can be found by subtracting the area of triangle ADE from the area of triangle ABC:
\[ \text{Area of DECB} = \text{Area of ABC} - \text{Area of ADE} = 4x - x = 3x \]

6. **Find the ratio of the area of triangle ABC to the area of quadrilateral DECB**:
\[ \text{Required ratio} = \frac{\text{Area of ABC}}{\text{Area of DECB}} = \frac{4x}{3x} = \frac{4}{3} \]

### Final Answer:
The ratio between the area of triangle ABC and the area of quadrilateral DECB is \( \frac{4}{3} \).