On a map drawn to 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 :

(i) the actual length of AB and BC in km

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

(i) Given,

Scale factor for reduction (k) = $\dfrac{1}{250000}$

⇒ Length of AB in model = k × Length of AB

⇒ 3 = $\dfrac{1}{250000}$ × Length of AB

⇒ Length of AB = 3 × 250000 = 750000 cm = $\dfrac{750000}{100 \times 1000}$ = 7.5 km.

⇒ Length of BC in model = k × Length of BC

⇒ 4 = $\dfrac{1}{250000}$ × Length of BC

⇒ Length of BC = 4 × 250000 = 1000000 cm = $\dfrac{1000000}{100 \times 1000}$ = 10 km.

Hence, actual length of AB = 7.5 km and BC = 10 km.

(ii) Area of plot = $\dfrac{1}{2} \times$ base × height

= $\dfrac{1}{2}$ × BC × AB

= $\dfrac{1}{2}$ × 10 × 7.5

= 37.5 sq. km.

Hence, area of plot = 37.5 sq. km.