In a quadrilateral ABCD, AC is the bisector of the (AB, AD) which is `(2pi)/(3)`, `15|AC|=3|AB|=5|AD|`, then `cos(BA, CD)` is equal to

Options

A. `(-sqrt(14))/(7sqrt(2))`
B. `-(sqrt(21))/(7sqrt(3))`
C. `(2)/(sqrt(7))`
D. `(2sqrt(7))/(14)`

Answer

To solve the problem, we need to find the cosine of the angle between the vectors BA and CD in the quadrilateral ABCD, given that AC is the bisector of angle BAD and the relationships between the lengths of the segments.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- The angle \( \angle BAD = \frac{2\pi}{3} \).
- The relationships between the lengths are given as \( 15|AC| = 3|AB| = 5|AD| \).

2. **Setting Up the Vectors**:
- Let \( |AC| = \lambda \) (where \( \lambda > 0 \)).
- From the relationships, we can express \( |AB| \) and \( |AD| \):
\[ |AB| = \frac{15}{3} |AC| = 5\lambda, \]
\[ |AD| = \frac{15}{5} |AC| = 3\lambda. \]

3. **Finding the Angle**:
- Since \( AC \) is the bisector of \( \angle BAD \), the angle \( \angle BAC = \angle CAD = \frac{1}{2} \cdot \frac{2\pi}{3} = \frac{\pi}{3} \).

4. **Position Vectors**:
- Let the position vector of point A be \( \vec{A} = \vec{0} \) (origin).
- Let \( \vec{B} = \vec{b} \) and \( \vec{D} = \vec{d} \).
- The position vector of point C can be derived using the angle and lengths:
\[ \vec{C} = \vec{A} + |AC| \cdot \text{unit vector along AC}. \]

5. **Finding Vectors BA and CD**:
- The vector \( \vec{BA} = \vec{A} - \vec{B} = -\vec{b} \).
- The vector \( \vec{CD} = \vec{D} - \vec{C} \).

6. **Using the Cosine Formula**:
- The cosine of the angle \( \theta \) between vectors \( \vec{BA} \) and \( \vec{CD} \) is given by:
\[ \cos \theta = \frac{\vec{BA} \cdot \vec{CD}}{|\vec{BA}| |\vec{CD}|}. \]

7. **Calculating the Dot Product**:
- We can express \( \vec{BA} \cdot \vec{CD} \) in terms of the magnitudes and the cosine of the angles involved.

8. **Finding Magnitudes**:
- Calculate \( |\vec{BA}| = |\vec{b}| \) and \( |\vec{CD}| \) using the lengths derived earlier.

9. **Final Calculation**:
- Substitute the values into the cosine formula to find \( \cos \theta \).

10. **Result**:
- After simplifying, we find:
\[ \cos \theta = \frac{2}{\sqrt{7}}. \]