IF a quadrilateral ABCD is such that `vecAB = b, vecAD = d` and `vecAC = pb + qd (p +q ge1)`. Then the area of he quadrilateral is `(1)/(2) (p +q) |bxxd|` . Is this statement true or false ?

Answer

To determine whether the statement about the area of quadrilateral ABCD is true or false, we will analyze the given vectors and apply the formula for the area of a quadrilateral.

### Step-by-Step Solution:

1. **Understanding the Vectors**:
- We have vectors defined as follows:
- \(\vec{AB} = \vec{b}\)
- \(\vec{AD} = \vec{d}\)
- \(\vec{AC} = p\vec{b} + q\vec{d}\) where \(p + q \geq 1\)

2. **Dividing the Quadrilateral**:
- The quadrilateral ABCD can be divided into two triangles: \(\triangle ABC\) and \(\triangle ACD\).

3. **Area of Triangle ABC**:
- The area of triangle ABC can be calculated using the formula:
\[ \text{Area}_{ABC} = \frac{1}{2} |\vec{AB} \times \vec{AC}| \]
- Substituting the vectors:
\[ \text{Area}_{ABC} = \frac{1}{2} |\vec{b} \times (p\vec{b} + q\vec{d})| \]
- Expanding this using the distributive property of the cross product:
\[ = \frac{1}{2} |\vec{b} \times p\vec{b} + \vec{b} \times q\vec{d}| \]
- Since \(\vec{b} \times \vec{b} = \vec{0}\), we have:
\[ \text{Area}_{ABC} = \frac{1}{2} |q(\vec{b} \times \vec{d})| \]
- Therefore:
\[ \text{Area}_{ABC} = \frac{q}{2} |\vec{b} \times \vec{d}| \]

4. **Area of Triangle ACD**:
- The area of triangle ACD can be calculated similarly:
\[ \text{Area}_{ACD} = \frac{1}{2} |\vec{AD} \times \vec{AC}| \]
- Substituting the vectors:
\[ \text{Area}_{ACD} = \frac{1}{2} |\vec{d} \times (p\vec{b} + q\vec{d})| \]
- Expanding this:
\[ = \frac{1}{2} |\vec{d} \times p\vec{b} + \vec{d} \times q\vec{d}| \]
- Again, since \(\vec{d} \times \vec{d} = \vec{0}\), we have:
\[ \text{Area}_{ACD} = \frac{p}{2} |\vec{d} \times \vec{b}| \]
- Therefore:
\[ \text{Area}_{ACD} = \frac{p}{2} |\vec{b} \times \vec{d}| \]

5. **Total Area of Quadrilateral ABCD**:
- The total area of quadrilateral ABCD is the sum of the areas of triangles ABC and ACD:
\[ \text{Area}_{ABCD} = \text{Area}_{ABC} + \text{Area}_{ACD} \]
- Substituting the areas we calculated:
\[ \text{Area}_{ABCD} = \frac{q}{2} |\vec{b} \times \vec{d}| + \frac{p}{2} |\vec{b} \times \vec{d}| \]
- Combining the terms:
\[ \text{Area}_{ABCD} = \frac{1}{2} (p + q) |\vec{b} \times \vec{d}| \]

6. **Conclusion**:
- The statement that the area of the quadrilateral is \(\frac{1}{2} (p + q) |\vec{b} \times \vec{d}|\) is **true**.