In the given figure, DE is parallel to BC and the ratio of the areas of `DeltaADE` and trapezium BDEC is 4:5. What is DE: BC?

Options

A. `1:2`
B. `2:3`
C. `4:5`
D. None of these

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step-by-Step Solution

1. **Understanding the Given Information**:
We have two parallel lines, DE and BC. The areas of triangle ADE and trapezium BDEC are given in the ratio of 4:5.

2. **Setting Up the Ratios**:
Let the area of triangle ADE be \( A_{ADE} = 4x \) and the area of trapezium BDEC be \( A_{BDEC} = 5x \).

3. **Finding the Total Area**:
The total area of triangle ABC can be expressed as:
\[ A_{ABC} = A_{ADE} + A_{BDEC} = 4x + 5x = 9x \]

4. **Using the Area Ratio of Similar Triangles**:
Since DE is parallel to BC, triangle ADE is similar to triangle ABC. The ratio of their areas is equal to the square of the ratio of their corresponding sides:
\[ \frac{A_{ADE}}{A_{ABC}} = \left(\frac{DE}{BC}\right)^2 \]
Substituting the areas we found:
\[ \frac{4x}{9x} = \left(\frac{DE}{BC}\right)^2 \]
Simplifying this gives:
\[ \frac{4}{9} = \left(\frac{DE}{BC}\right)^2 \]

5. **Taking the Square Root**:
To find the ratio of DE to BC, we take the square root of both sides:
\[ \frac{DE}{BC} = \sqrt{\frac{4}{9}} = \frac{2}{3} \]

6. **Final Ratio**:
Thus, the ratio of DE to BC is:
\[ DE : BC = 2 : 3 \]