Point D and E respectively lie on the sides AB and AC of a triangle such that `DE||BC`. If area of `DeltaABC` and trapezium DECB are in the ratio `16:15`, then `AE:AC` is

Options

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

Answer

To solve the problem, we need to find the ratio \( AE:EC \) given that \( DE \parallel BC \) and the areas of triangle \( ABC \) and trapezium \( DECB \) are in the ratio \( 16:15 \).

### Step-by-Step Solution:

1. **Understand the Given Information**:
We have triangle \( ABC \) with points \( D \) and \( E \) on sides \( AB \) and \( AC \) respectively, such that \( DE \parallel BC \). The areas of triangle \( ABC \) and trapezium \( DECB \) are in the ratio \( 16:15 \).

2. **Identify the Areas**:
Let the area of triangle \( ABC \) be \( 16x \) and the area of trapezium \( DECB \) be \( 15x \). The total area of triangle \( ABC \) can be expressed as the sum of the area of triangle \( ADE \) and the area of trapezium \( DECB \):
\[ \text{Area of } ABC = \text{Area of } ADE + \text{Area of } DECB \]
Thus, we have:
\[ 16x = \text{Area of } ADE + 15x \]
This simplifies to:
\[ \text{Area of } ADE = 16x - 15x = x \]

3. **Use the Area Ratio to Find Side Ratio**:
Since \( DE \parallel BC \), the triangles \( ADE \) and \( ABC \) are similar. The ratio of their areas is equal to the square of the ratio of their corresponding sides. Therefore, we have:
\[ \frac{\text{Area of } ADE}{\text{Area of } ABC} = \left(\frac{AE}{AC}\right)^2 \]
Substituting the areas we found:
\[ \frac{x}{16x} = \left(\frac{AE}{AC}\right)^2 \]
This simplifies to:
\[ \frac{1}{16} = \left(\frac{AE}{AC}\right)^2 \]

4. **Calculate the Side Ratio**:
Taking the square root of both sides gives:
\[ \frac{AE}{AC} = \frac{1}{4} \]

5. **Find the Ratio \( AE:EC \)**:
Since \( AC = AE + EC \), we can express \( EC \) in terms of \( AE \):
\[ EC = AC - AE = 4k - k = 3k \quad \text{(where \( AE = k \))} \]
Thus, the ratio \( AE:EC \) is:
\[ AE:EC = k:3k = 1:3 \]

### Final Answer:
\[ AE:EC = 1:3 \]