In a rhombus ABCD, `angleADC:angleBCD=1:2 and EF:BD=1:3`, where E and F lie on the diagonal BD, inside the rhombus, such that `square AECF` is a quadrlateral. Find the ratio of areas of `squareAECF: square ABCD`.

Options

A. A. `1:4`
B. B. `2:5`
C. c. `3:14`
D. D. none of these

Answer

To solve the problem, we need to find the ratio of the areas of quadrilateral AECF to quadrilateral ABCD in the rhombus ABCD, given the conditions about angles and segments.

### Step-by-Step Solution:

1. **Understanding the Rhombus**:
- In a rhombus, all sides are equal, and opposite angles are equal. We have angles ADC and BCD in the ratio of 1:2. Let's denote angle ADC as \( x \) and angle BCD as \( 2x \).

2. **Finding Angles**:
- The sum of angles in a quadrilateral is \( 360^\circ \). Therefore, we can write:
\[ x + 2x + x + 2x = 360^\circ \]
This simplifies to:
\[ 6x = 360^\circ \implies x = 60^\circ \]
- Thus, angle ADC = \( 60^\circ \) and angle BCD = \( 120^\circ \).

3. **Understanding the Segment Ratios**:
- We are given that \( EF : BD = 1 : 3 \). If we let \( EF = k \), then \( BD = 3k \).

4. **Area of Rhombus ABCD**:
- The area of a rhombus can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals. Here, we can assume \( d_1 = EF = k \) and \( d_2 = BD = 3k \).
\[ \text{Area of ABCD} = \frac{1}{2} \times k \times 3k = \frac{3k^2}{2} \]

5. **Area of Quadrilateral AECF**:
- The area of quadrilateral AECF can also be expressed in terms of its diagonal \( EF \) and the other diagonal \( AC \). Since \( EF = k \) and \( AC \) can be assumed to be equal to \( BD \) (as diagonals bisect each other in a rhombus), we can use the same diagonal for calculation.
- The area of AECF can be calculated as:
\[ \text{Area of AECF} = \frac{1}{2} \times EF \times AC = \frac{1}{2} \times k \times k = \frac{k^2}{2} \]

6. **Finding the Ratio of Areas**:
- Now we can find the ratio of the areas of quadrilateral AECF to quadrilateral ABCD:
\[ \text{Ratio} = \frac{\text{Area of AECF}}{\text{Area of ABCD}} = \frac{\frac{k^2}{2}}{\frac{3k^2}{2}} = \frac{1}{3} \]

### Final Answer:
The ratio of areas of quadrilateral AECF to quadrilateral ABCD is \( \frac{1}{3} \).