On a map drawn to a scale of 1 : 250000, a triangular plot of land has the following measurements :

AB = 3 cm, BC = 4 cm and ∠ABC = 90°. Calculate :

(i) the actual length of AB in km.

(ii) the area of the plot in sq. km.

(i) Since, the model of the triangular plot is made to the scale of 1 : 250000.

∴ K (Scale factor) = 250000.

Actual length of AB = k × (the length of AB in model) = 250000 × 3 = 750000 cm.

1 cm = 10^-5 km.

∴ 750000 cm = 750000 × 10^-5 km = 7.5 km.

Hence, the length of AB is 7.5 km.

(ii) Since the plot is a right angled triangle.

We know area of right angled triangle is given by

$\Rightarrow \dfrac{1}{2} \times \text{ base } \times \text{ height}.$

Hence, area of the model is,

$= \dfrac{1}{2} \times AB \times BC \\[1em] = \dfrac{1}{2} \times 3 \times 4 \\[1em] = 6 \text{ cm}^2.$

We know that 1 cm² = 10^-10 km².

Hence, area of model = 6 × 10^-10 km².

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

Putting values in above equation,

$= (250000)^2 \times 6 \times 10^{-10} \\[1em] = 625 \times 10^{8} \times 6 \times 10^{-10} \\[1em] = 3750 \times 10^{-2.} \\[1em] = 37.5 \text{ km}^2.$

Hence, the area of the plot is 37.5 km².