The scale of map is 1 : 50,000. In the map, a triangular plot ABC of land has the following dimensions :
`AB = 2 cm, BC = 3-5 cm ` and angle `ABC = 90^@`.
Calculate : the area of the plot in sq. km.

Answer

To solve the problem, we need to calculate the area of the triangular plot ABC based on the dimensions provided on the map and the scale of the map.

### Step-by-Step Solution:

1. **Understand the Scale of the Map**:
The scale of the map is given as 1 : 50,000. This means that 1 cm on the map represents 50,000 cm in reality.

2. **Identify the Dimensions on the Map**:
We have the following dimensions on the map:
- AB = 2 cm
- BC = 3.5 cm
- Angle ABC = 90° (indicating that triangle ABC is a right triangle).

3. **Calculate the Actual Lengths**:
To find the actual lengths of AB and BC, we multiply the lengths on the map by the scale factor (50,000).

- Actual length of AB:
\[ \text{Actual length of AB} = 2 \, \text{cm} \times 50,000 = 100,000 \, \text{cm} \]
Converting centimeters to kilometers:
\[ 100,000 \, \text{cm} = 1 \, \text{km} \]

- Actual length of BC:
\[ \text{Actual length of BC} = 3.5 \, \text{cm} \times 50,000 = 175,000 \, \text{cm} \]
Converting centimeters to kilometers:
\[ 175,000 \, \text{cm} = 1.75 \, \text{km} \]

4. **Calculate the Area of the Triangle**:
The area \(A\) of a right triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take AB as the base and BC as the height:
\[ A = \frac{1}{2} \times 1 \, \text{km} \times 1.75 \, \text{km} \]
\[ A = \frac{1}{2} \times 1.75 = 0.875 \, \text{sq. km} \]

5. **Final Answer**:
The area of the triangular plot ABC is \(0.875 \, \text{sq. km}\).