On a map drawn to a scale of `1:50,000`, a rectangular plot of alnd ABCD has the following dimensions. AB = 6 cm, BC = 8 cm and all angles are right angles. Find :
(i) the actual length of the diagonal distyance AC of the plot in km.

Answer

To find the actual length of the diagonal distance AC of the rectangular plot of land ABCD on a map drawn to a scale of 1:50,000, we can follow these steps:

### Step 1: Identify the dimensions of the rectangle
Given:
- AB = 6 cm
- BC = 8 cm

### Step 2: Use the Pythagorean theorem to find the length of diagonal AC
Since ABCD is a rectangle, we can use the Pythagorean theorem to find the length of the diagonal AC.

\[ AC^2 = AB^2 + BC^2 \]

Substituting the values:

\[ AC^2 = (6 \, \text{cm})^2 + (8 \, \text{cm})^2 \]
\[ AC^2 = 36 \, \text{cm}^2 + 64 \, \text{cm}^2 \]
\[ AC^2 = 100 \, \text{cm}^2 \]
\[ AC = \sqrt{100 \, \text{cm}^2} = 10 \, \text{cm} \]

### Step 3: Convert the length from centimeters to actual distance using the scale
The scale of the map is 1:50,000, which means that 1 cm on the map represents 50,000 cm in reality.

To find the actual length of AC:

\[ \text{Actual length of AC} = 10 \, \text{cm} \times 50,000 = 500,000 \, \text{cm} \]

### Step 4: Convert centimeters to kilometers
To convert centimeters to kilometers, we use the conversion factor:

\[ 1 \, \text{km} = 100,000 \, \text{cm} \]

Thus, we convert the actual length:

\[ \text{Actual length of AC in km} = \frac{500,000 \, \text{cm}}{100,000 \, \text{cm/km}} = 5 \, \text{km} \]

### Final Answer
The actual length of the diagonal distance AC of the plot is **5 km**.

---