Find the area of a quadrilateral one of whose diagonals is 30 cm long and the perpendiculars from the other two vertices are 19 cm and 11 cm respectively.

Answer

To find the area of the quadrilateral with the given dimensions, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Length of diagonal (d) = 30 cm
- Perpendicular from one vertex (h1) = 19 cm
- Perpendicular from the other vertex (h2) = 11 cm

2. **Use the Area Formula for a Quadrilateral**:
The area (A) of a quadrilateral when one diagonal and the perpendiculars from the opposite vertices are known can be calculated using the formula:
\[ A = \frac{1}{2} \times d \times (h1 + h2) \]

3. **Substitute the Values into the Formula**:
Substitute the values of d, h1, and h2 into the formula:
\[ A = \frac{1}{2} \times 30 \times (19 + 11) \]

4. **Calculate the Sum of the Perpendiculars**:
Calculate \(h1 + h2\):
\[ 19 + 11 = 30 \]

5. **Substitute the Sum Back into the Area Formula**:
Now substitute this sum back into the area formula:
\[ A = \frac{1}{2} \times 30 \times 30 \]

6. **Perform the Multiplication**:
Calculate \(30 \times 30\):
\[ 30 \times 30 = 900 \]

7. **Calculate the Final Area**:
Now calculate \(\frac{1}{2} \times 900\):
\[ A = \frac{900}{2} = 450 \text{ cm}^2 \]

### Final Answer:
The area of the quadrilateral is \(450 \text{ cm}^2\).

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