D and E are points on sides AB and AC of `triangleABC` DE is parallel to BC if AD: DB =1 :2 and area of `triangleABC` is 45 sq cm what is the area of quadrialteral BDEC

Options

A. 20
B. 40
C. 15
D. 30

Answer

To solve the problem, we need to find the area of quadrilateral BDEC in triangle ABC, given that DE is parallel to BC, AD:DB = 1:2, and the area of triangle ABC is 45 sq cm.

### Step-by-Step Solution:

1. **Understanding the Ratios**:
- We have AD:DB = 1:2. This means if we let AD = 1x, then DB = 2x. Therefore, AB (which is AD + DB) = 1x + 2x = 3x.

2. **Finding the Area of Triangle ADE**:
- Since DE is parallel to BC, triangles ADE and ABC are similar by the Basic Proportionality Theorem (also known as Thales' theorem).
- The ratio of the areas of similar triangles is equal to the square of the ratio of their corresponding sides.
- The ratio of the sides AD to AB is given by:
\[ \frac{AD}{AB} = \frac{1}{3} \]
- Therefore, the ratio of the areas of triangle ADE to triangle ABC is:
\[ \left(\frac{AD}{AB}\right)^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]

3. **Calculating the Area of Triangle ADE**:
- The area of triangle ABC is given as 45 sq cm.
- Using the area ratio, we can find the area of triangle ADE:
\[ \text{Area of } \triangle ADE = \text{Area of } \triangle ABC \times \frac{1}{9} = 45 \times \frac{1}{9} = 5 \text{ sq cm} \]

4. **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:
\[ \text{Area of } BDEC = \text{Area of } \triangle ABC - \text{Area of } \triangle ADE = 45 - 5 = 40 \text{ sq cm} \]

### Final Answer:
The area of quadrilateral BDEC is **40 sq cm**.