Find the area of quadrilateral ABCD whose diagonal AC is 10 cm long and the lengths of perpendicular drawn from the vertices B and D on AC are 4 cm and 3 cm , respectively .

Answer

To find the area of quadrilateral ABCD, we can use the formula for the area of a quadrilateral when the lengths of the diagonals and the perpendicular heights from the opposite vertices are known.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Length of diagonal AC (d) = 10 cm
- Length of perpendicular from vertex B to AC (h1) = 4 cm
- Length of perpendicular from vertex D to AC (h2) = 3 cm

2. **Calculate the Area of Triangles:**
- The area of triangle ABC can be calculated using the formula:
\[ \text{Area of } \triangle ABC = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is AC and the height is the perpendicular from B.
\[ \text{Area of } \triangle ABC = \frac{1}{2} \times 10 \times 4 = 20 \text{ cm}^2 \]

- Similarly, the area of triangle ADC can be calculated:
\[ \text{Area of } \triangle ADC = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is AC and the height is the perpendicular from D.
\[ \text{Area of } \triangle ADC = \frac{1}{2} \times 10 \times 3 = 15 \text{ cm}^2 \]

3. **Calculate the Total Area of Quadrilateral ABCD:**
- The area of quadrilateral ABCD is the sum of the areas of triangles ABC and ADC:
\[ \text{Area of quadrilateral ABCD} = \text{Area of } \triangle ABC + \text{Area of } \triangle ADC \]
\[ \text{Area of quadrilateral ABCD} = 20 + 15 = 35 \text{ cm}^2 \]

### Final Answer:
The area of quadrilateral ABCD is **35 cm²**.