ABCD is a quadrilateral in which diagonal BD = 64 cm. `AL bot BD" and "CM bot BD`, such that `AL=13.2cm" and "CM=16.8cm`. The area of the quadrilateral ABCD is square centimeters is :

Options

A. a) 422.4
B. b) 690.0
C. c) 537.6
D. d) 960.0

Answer

To find the area of quadrilateral ABCD, we can break it down into two triangles: triangle ABD and triangle BCD. The area of a triangle can be calculated using the formula:

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Diagonal \( BD = 64 \, \text{cm} \)
- Height from point \( A \) to line \( BD \) (denoted as \( AL \)) = \( 13.2 \, \text{cm} \)
- Height from point \( C \) to line \( BD \) (denoted as \( CM \)) = \( 16.8 \, \text{cm} \)

2. **Calculate the Area of Triangle ABD:**
- For triangle ABD, the base is \( BD \) and the height is \( AL \).
- Using the area formula:
\[ \text{Area of } \triangle ABD = \frac{1}{2} \times BD \times AL = \frac{1}{2} \times 64 \times 13.2 \]

3. **Calculate the Area of Triangle BCD:**
- For triangle BCD, the base is also \( BD \) and the height is \( CM \).
- Using the area formula:
\[ \text{Area of } \triangle BCD = \frac{1}{2} \times BD \times CM = \frac{1}{2} \times 64 \times 16.8 \]

4. **Sum the Areas of Both Triangles:**
- The total area of quadrilateral ABCD is the sum of the areas of triangles ABD and BCD:
\[ \text{Area of } ABCD = \text{Area of } \triangle ABD + \text{Area of } \triangle BCD \]

5. **Substituting the Values:**
- Calculate the area of triangle ABD:
\[ \text{Area of } \triangle ABD = \frac{1}{2} \times 64 \times 13.2 = 32 \times 13.2 = 422.4 \, \text{cm}^2 \]
- Calculate the area of triangle BCD:
\[ \text{Area of } \triangle BCD = \frac{1}{2} \times 64 \times 16.8 = 32 \times 16.8 = 537.6 \, \text{cm}^2 \]

6. **Final Calculation:**
- Now, sum the areas:
\[ \text{Area of } ABCD = 422.4 + 537.6 = 960 \, \text{cm}^2 \]

### Conclusion:
The area of quadrilateral ABCD is \( 960 \, \text{cm}^2 \).