ABCD is a quadrilateral in which BD = 40 cm . The lenghtsof the perpendiculars drawn from the opposite vertices on BD are 16 cm and 12 cm . The area of the quadrilateral in (in `cm^(2)` ) is

Options

A. 560
B. 530
C. 800
D. 500

Answer

To find the area of the quadrilateral ABCD, we can use the formula for the area of a trapezium (trapezoid) since we have two triangles formed by the diagonals. The area of a quadrilateral can be calculated using the lengths of the diagonals and the heights from the opposite vertices.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Length of diagonal BD = 40 cm
- Length of the perpendicular from vertex A to line BD = 16 cm
- Length of the perpendicular from vertex C to line BD = 12 cm

2. **Calculate the Area of Triangle ABD:**
The area of triangle ABD can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is BD and the height is the perpendicular from A.
\[ \text{Area of } \triangle ABD = \frac{1}{2} \times 40 \, \text{cm} \times 16 \, \text{cm} = \frac{1}{2} \times 640 \, \text{cm}^2 = 320 \, \text{cm}^2 \]

3. **Calculate the Area of Triangle BCD:**
Similarly, we can calculate the area of triangle BCD using the same formula:
\[ \text{Area of } \triangle BCD = \frac{1}{2} \times 40 \, \text{cm} \times 12 \, \text{cm} = \frac{1}{2} \times 480 \, \text{cm}^2 = 240 \, \text{cm}^2 \]

4. **Calculate the Total Area of Quadrilateral ABCD:**
The total area of quadrilateral ABCD is the sum of the areas of triangles ABD and BCD:
\[ \text{Area of quadrilateral ABCD} = \text{Area of } \triangle ABD + \text{Area of } \triangle BCD \]
\[ \text{Area of quadrilateral ABCD} = 320 \, \text{cm}^2 + 240 \, \text{cm}^2 = 560 \, \text{cm}^2 \]

### Final Answer:
The area of the quadrilateral ABCD is **560 cm²**.