The diagonal of a quadrilateral shaped field is 24 m and the perpendiculars dropped on its from the remaining opposite vertices are 8m and 13m. The area of the field is

Options

A. `252 m^(2)`
B. `156 m^(2)`
C. `96 m^(2)`
D. `1152 m^(2)`

Answer

To find the area of the quadrilateral-shaped field, we can divide it into two triangles using the diagonal. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Length of the diagonal (AC) = 24 m
- Perpendicular dropped from vertex B = 8 m
- Perpendicular dropped from vertex D = 13 m

### Step 2: Divide the quadrilateral into two triangles
We can divide the quadrilateral ABCD into two triangles:
1. Triangle ACD
2. Triangle ABC

### Step 3: Calculate the area of triangle ACD
The area of a triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For triangle ACD:
- Base (AC) = 24 m
- Height (from D) = 13 m

Substituting the values:
\[ \text{Area of triangle ACD} = \frac{1}{2} \times 24 \times 13 \]
\[ = 12 \times 13 = 156 \text{ m}^2 \]

### Step 4: Calculate the area of triangle ABC
For triangle ABC:
- Base (AC) = 24 m
- Height (from B) = 8 m

Substituting the values:
\[ \text{Area of triangle ABC} = \frac{1}{2} \times 24 \times 8 \]
\[ = 12 \times 8 = 96 \text{ m}^2 \]

### Step 5: Calculate the total area of the quadrilateral
Now, we add the areas of both triangles to get the total area of the quadrilateral:
\[ \text{Total Area} = \text{Area of triangle ACD} + \text{Area of triangle ABC} \]
\[ = 156 \text{ m}^2 + 96 \text{ m}^2 = 252 \text{ m}^2 \]

### Final Answer
The area of the field is **252 m²**.

---