In a four sided figure , the longer diagonal is 16cm. The perpendiculars from the opposite vertices falling upon the longer diagonal are 10 cm and 12 cm . What is the area of the quadrilateral ?

Options

A. ` 170 cm ^(2)`
B. ` 172 cm ^(2)`
C. ` 174 cm ^(2)`
D. ` 176 cm ^(2)`

Answer

To find the area of the quadrilateral given the longer diagonal and the perpendicular distances from the opposite vertices, we can use the formula for the area of a quadrilateral based on its diagonals and the perpendiculars dropped from the opposite vertices.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- The length of the longer diagonal (L) = 16 cm
- The perpendicular distance from one vertex to the diagonal (D1) = 10 cm
- The perpendicular distance from the opposite vertex to the diagonal (D2) = 12 cm

2. **Use the Area Formula**:
The area (A) of the quadrilateral can be calculated using the formula:
\[ A = \frac{1}{2} \times L \times (D1 + D2) \]
where \(L\) is the length of the diagonal and \(D1\) and \(D2\) are the perpendicular distances from the opposite vertices.

3. **Substitute the Values**:
Substitute the known values into the formula:
\[ A = \frac{1}{2} \times 16 \times (10 + 12) \]

4. **Calculate the Sum of Perpendiculars**:
Calculate \(D1 + D2\):
\[ D1 + D2 = 10 + 12 = 22 \]

5. **Calculate the Area**:
Now substitute this back into the area formula:
\[ A = \frac{1}{2} \times 16 \times 22 \]
\[ A = 8 \times 22 = 176 \text{ cm}^2 \]

6. **Final Result**:
Therefore, the area of the quadrilateral is:
\[ \text{Area} = 176 \text{ cm}^2 \]