The distance between the two parallel lines is 1 unit. A point A is chosen to lie between the lines at a distance 'd' from one of them Triangle ABC is equilateral with B on one line and C on the other parallel line. The length of the side of the equilateral triangle is

Options

A. `2/3 sqrt(d^(2)+d+1)`
B. `2sqrt((d^(2)-d+1)/(3))`
C. `2sqrt(d^(2)-d+1)`
D. `sqrt(d^(2)-d+1)`

Answer

To find the length of the side of the equilateral triangle ABC, we will follow a systematic approach using the given information about the distance between the two parallel lines and the position of point A.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Let the two parallel lines be L1 and L2, with a distance of 1 unit between them.
- Point A is located at a distance \( d \) from line L1.
- Therefore, the distance from point A to line L2 will be \( 1 - d \).
- Points B and C are on lines L1 and L2, respectively.

2. **Setting Up the Triangle**:
- Since triangle ABC is equilateral, all sides are equal. Let the length of each side be \( s \).
- The height of the equilateral triangle can be expressed in terms of \( s \) as \( \frac{\sqrt{3}}{2} s \).

3. **Using the Height of the Triangle**:
- The height of triangle ABC can also be represented as the distance from point A to line L2 plus the distance from point A to line L1.
- Therefore, the height of triangle ABC can be expressed as:
\[ \text{Height} = d + (1 - d) = 1 \]
- This means that the height of the triangle is equal to 1 unit.

4. **Relating Height to Side Length**:
- For an equilateral triangle, the height \( h \) is related to the side length \( s \) by the formula:
\[ h = \frac{\sqrt{3}}{2} s \]
- Setting this equal to the height we found:
\[ \frac{\sqrt{3}}{2} s = 1 \]

5. **Solving for Side Length \( s \)**:
- To find \( s \), we can rearrange the equation:
\[ s = \frac{2}{\sqrt{3}} \]

6. **Rationalizing the Denominator**:
- To express \( s \) in a more standard form, we can rationalize the denominator:
\[ s = \frac{2 \sqrt{3}}{3} \]

### Final Answer:
The length of the side of the equilateral triangle ABC is:
\[ s = \frac{2 \sqrt{3}}{3} \]