ABC is a given triangle. A straight line EF is drawn parallel to BC. It cuts AB at E and AC at F. If area of AEF is one third area of quadrilateral EBCF then `EB:AB` is

Options

A. `1:2`
B. `1:sqrt(2)`
C. `1:sqrt(3)`
D. `1:9`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Given Information
We have triangle ABC, and a line EF is drawn parallel to BC, intersecting AB at E and AC at F. The area of triangle AEF is given to be one-third of the area of quadrilateral EBCF.

### Step 2: Set Up the Relationship Between Areas
Let the area of triangle AEF be denoted as \( A_{AEF} \) and the area of quadrilateral EBCF as \( A_{EBCF} \). According to the problem, we have:
\[ A_{AEF} = \frac{1}{3} A_{EBCF} \]

### Step 3: Express the Area of Quadrilateral EBCF
The area of quadrilateral EBCF can be expressed as:
\[ A_{EBCF} = A_{ABC} - A_{AEF} \]
where \( A_{ABC} \) is the area of triangle ABC.

### Step 4: Substitute the Area Relationship
Substituting the expression for \( A_{EBCF} \) into the area relationship gives us:
\[ A_{AEF} = \frac{1}{3} \left( A_{ABC} - A_{AEF} \right) \]

### Step 5: Solve for \( A_{AEF} \)
Multiplying both sides by 3 to eliminate the fraction:
\[ 3A_{AEF} = A_{ABC} - A_{AEF} \]
Adding \( A_{AEF} \) to both sides:
\[ 4A_{AEF} = A_{ABC} \]
Thus, we can express the area of triangle AEF in terms of the area of triangle ABC:
\[ A_{AEF} = \frac{1}{4} A_{ABC} \]

### Step 6: Use the Similarity of Triangles
Since EF is parallel to BC, triangles AEF and ABC are similar by the Angle-Angle (AA) criterion. Therefore, the ratio of their areas is equal to the square of the ratio of their corresponding sides:
\[ \frac{A_{AEF}}{A_{ABC}} = \left( \frac{AE}{AB} \right)^2 \]
Substituting the area relationship:
\[ \frac{1}{4} = \left( \frac{AE}{AB} \right)^2 \]

### Step 7: Solve for the Ratio of Sides
Taking the square root of both sides:
\[ \frac{AE}{AB} = \frac{1}{2} \]
This implies:
\[ \frac{EB}{AB} = 1 - \frac{AE}{AB} = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 8: Final Ratio
Thus, the ratio \( EB:AB \) is:
\[ EB:AB = 1:2 \]

### Conclusion
The final answer is \( EB:AB = 1:2 \).

---