In `DeltaABC`, D and E are points on AB and AC respectively such that DE||BC and DE divides the `DeltaABC` into two parts of equal areas. Then ratio of AD and BD is

Options

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

Answer

To solve the problem step by step, we will analyze the given triangle \( \Delta ABC \) and the points \( D \) and \( E \) on sides \( AB \) and \( AC \) respectively, where \( DE \parallel BC \) and \( DE \) divides the triangle into two equal areas.

### Step 1: Understand the similarity of triangles
Since \( DE \parallel BC \), triangles \( \Delta ADE \) and \( \Delta ABC \) are similar by the AA (Angle-Angle) similarity criterion. This means that the corresponding angles are equal.

**Hint:** Remember that parallel lines create corresponding angles that are equal.

### Step 2: Set up the area relationship
Let the area of triangle \( ADE \) be \( x \). Since \( DE \) divides the triangle \( ABC \) into two equal areas, the area of triangle \( ABC \) will be \( 2x \).

**Hint:** The area of the larger triangle is the sum of the areas of the smaller triangle and the trapezium formed.

### Step 3: Use the ratio of areas to find the ratio of sides
From the similarity of triangles, we know that:
\[ \frac{\text{Area of } \Delta ADE}{\text{Area of } \Delta ABC} = \left(\frac{AD}{AB}\right)^2 \]
Substituting the areas we have:
\[ \frac{x}{2x} = \left(\frac{AD}{AB}\right)^2 \]
This simplifies to:
\[ \frac{1}{2} = \left(\frac{AD}{AB}\right)^2 \]

**Hint:** The ratio of the areas of similar triangles is equal to the square of the ratio of their corresponding sides.

### Step 4: Solve for the ratio of sides
Taking the square root of both sides gives us:
\[ \frac{AD}{AB} = \frac{1}{\sqrt{2}} \]
Let \( AD = k \). Then:
\[ AB = \sqrt{2}k \]

**Hint:** When working with ratios, it can be helpful to express one variable in terms of another.

### Step 5: Find \( BD \)
Since \( AB = AD + BD \), we can express \( BD \) as follows:
\[ BD = AB - AD = \sqrt{2}k - k = (\sqrt{2} - 1)k \]

**Hint:** Always remember to express the total length in terms of the parts you have defined.

### Step 6: Calculate the ratio \( \frac{AD}{BD} \)
Now we can find the ratio \( \frac{AD}{BD} \):
\[ \frac{AD}{BD} = \frac{k}{(\sqrt{2} - 1)k} = \frac{1}{\sqrt{2} - 1} \]

To simplify, we can rationalize the denominator:
\[ \frac{1}{\sqrt{2} - 1} \cdot \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{\sqrt{2} + 1}{1} = \sqrt{2} + 1 \]

Thus, the ratio \( AD : BD \) is \( 1 : (\sqrt{2} - 1) \).

### Final Answer
The ratio of \( AD \) to \( BD \) is \( 1 : (\sqrt{2} - 1) \).

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