A map of a square plot of land is drawn to a scale 1 : 25000. If the area of the plot in the map is 72 cm², find :

(i) the actual area of the plot of land.

(ii) the length of the diagonal in the actual plot of land.

Hint : (ii) $\dfrac{1}{2}$ (length of diagonal)² = area of square.

(i) Since, the model of the square plot is constructed with scale 1 : 25000.

k (Scale factor) = 25000.

Area of the actual plot = k² × (Area of the model of the plot)

Given, area of the model = 72 cm². Putting values in above equation,

$= (25000)^2 \times 72 \\[1em] = 625000000 \times 72 \\[1em] = 45000000000 \\[1em] = 45 \times 10^9 \text{ cm}^2.$

We know that 1 cm² = 10^-10 km².

∴ Actual area of plot = 45 × 10⁹ × 10^-10 km² = 4.5 km².

Hence, the actual area of the plot is 4.5 km².

(ii) We know that,

$\dfrac{1}{2}$ (length of diagonal)² = area of square.

Putting value of area of square plot = 4.5 km² in above equation we get,

$\Rightarrow \dfrac{1}{2} \text{ (Length of diagonal)}^2 = 4.5 \\[1em] \Rightarrow \text{(Length of diagonal)}^2 = 9 \\[1em] \Rightarrow \text{Length of diagonal} = \sqrt{9} \\[1em] \Rightarrow \text{ Length of diagonal} = 3.$

Hence, the length of diagonal in the actual plot of land is 3 km.