On a map drawn to a scale of 1 : 2,50,000, a triangular plot of land has the following measurements: `AB = 3 cm, BC = 4 cm `and angle `ABC = 90^@` Calculate :
the actual lengths of AB and BC in km.

Answer

To solve the problem, we need to calculate the actual lengths of sides AB and BC of a triangular plot of land based on the measurements given on a map with a scale of 1:250,000.

### Step-by-Step Solution:

1. **Identify the Measurements:**
- From the problem, we have:
- \( AB = 3 \, \text{cm} \)
- \( BC = 4 \, \text{cm} \)
- \( \angle ABC = 90^\circ \)

2. **Understanding the Scale:**
- The scale of the map is 1:250,000, which means 1 cm on the map represents 250,000 cm in reality.

3. **Calculate the Actual Length of AB:**
- To find the actual length of \( AB \):
\[ \text{Actual length of } AB = AB \times \text{Scale} = 3 \, \text{cm} \times 250,000 \, \text{cm} \]
\[ \text{Actual length of } AB = 750,000 \, \text{cm} \]

4. **Convert the Length from Centimeters to Kilometers:**
- We know that:
- 1 km = 100,000 cm
- To convert \( 750,000 \, \text{cm} \) to kilometers:
\[ \text{Length in km} = \frac{750,000 \, \text{cm}}{100,000 \, \text{cm/km}} = 7.5 \, \text{km} \]

5. **Calculate the Actual Length of BC:**
- To find the actual length of \( BC \):
\[ \text{Actual length of } BC = BC \times \text{Scale} = 4 \, \text{cm} \times 250,000 \, \text{cm} \]
\[ \text{Actual length of } BC = 1,000,000 \, \text{cm} \]

6. **Convert the Length from Centimeters to Kilometers:**
- To convert \( 1,000,000 \, \text{cm} \) to kilometers:
\[ \text{Length in km} = \frac{1,000,000 \, \text{cm}}{100,000 \, \text{cm/km}} = 10 \, \text{km} \]

### Final Results:
- The actual length of \( AB \) is **7.5 km**.
- The actual length of \( BC \) is **10 km**.