In `DeltaABC`, XY is parallel to BC and it divides the triangle into two equal areas (X lies on AB and Y lies on AC) then `BX:AB` equals

Options

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

Answer

To solve the problem, we need to find the ratio \( BX:AB \) in triangle \( ABC \) where line \( XY \) is parallel to \( BC \) and divides the triangle into two equal areas. Here’s a step-by-step solution:

### Step 1: Understand the Geometry
In triangle \( ABC \), line \( XY \) is drawn parallel to side \( BC \), intersecting \( AB \) at point \( X \) and \( AC \) at point \( Y \). Since \( XY \) is parallel to \( BC \), triangles \( AXY \) and \( ABC \) are similar.

**Hint:** Remember that when two lines are parallel, corresponding angles are equal, which implies similarity of triangles.

### Step 2: Set Up Area Relationships
Since \( XY \) divides triangle \( ABC \) into two equal areas, we can denote the area of triangle \( AXY \) as \( 1 \) unit and the area of triangle \( ABC \) as \( 2 \) units.

**Hint:** The total area of triangle \( ABC \) is the sum of the areas of triangles \( AXY \) and \( XYB \).

### Step 3: Use the Area Ratio to Find Side Ratios
The ratio of the areas of similar triangles is equal to the square of the ratio of their corresponding sides. Thus, we have:

\[ \frac{Area(ABC)}{Area(AXY)} = \frac{2}{1} = 2 \]

This implies:

\[ \left(\frac{AB}{AX}\right)^2 = 2 \]

Taking the square root gives:

\[ \frac{AB}{AX} = \sqrt{2} \]

**Hint:** When dealing with similar triangles, remember that the ratio of areas relates to the square of the ratio of their corresponding sides.

### Step 4: Express \( BX \) in Terms of \( AB \)
Let \( BX = x \) and \( AX = AB - x \). From the ratio we found:

\[ \frac{AB}{AX} = \sqrt{2} \implies AX = \frac{AB}{\sqrt{2}} \]

Now substituting \( AX \):

\[ AB - x = \frac{AB}{\sqrt{2}} \]

Rearranging gives:

\[ x = AB - \frac{AB}{\sqrt{2}} = AB \left(1 - \frac{1}{\sqrt{2}}\right) \]

**Hint:** Rearranging equations can help isolate the variable you need.

### Step 5: Find the Ratio \( BX:AB \)
Now we can express the ratio \( BX:AB \):

\[ BX = AB \left(1 - \frac{1}{\sqrt{2}}\right) \]

Thus, the ratio \( BX:AB \) becomes:

\[ BX:AB = \left(1 - \frac{1}{\sqrt{2}}\right):1 \]

**Hint:** When finding ratios, ensure both parts of the ratio are in terms of the same variable.

### Final Answer
The ratio \( BX:AB \) is:

\[ BX:AB = \left(1 - \frac{1}{\sqrt{2}}\right):1 \]

This can be simplified further, but the essential form is established.