On a map, drawn to a scale of 1 : 20000, a rectangular plot of land ABCD has AB = 24 cm and BC = 32 cm. Calculate :

(i) the diagonal distance of the plot in kilometre.

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

The rectangular plot of land ABCD on the map is shown below:

(i) By pythagoras theorem,

⇒ AC² = AB² + BC²

⇒ AC² = 24² + 32²

⇒ AC² = 576 + 1024

⇒ AC² = 1600

⇒ AC = $\sqrt{1600}$ = 40 cm.

Given,

Scale (k) = $\dfrac{1}{20000}$

Length of diagonal distance in map = k × Length of diagonal of plot

40 = $\dfrac{1}{20000}$ × Length of diagonal of plot

Length of diagonal of plot = 40 × 20000 = 800000 cm = $\dfrac{800000}{100000}$ km = 8 km.

Hence, diagonal distance of plot = 8 km.

(ii) Area of plot on map = AB × BC = 24 × 32 = 768 cm².

Area of plot on map = k² × Area of actual plot

768 = $\Big(\dfrac{1}{20000}\Big)^2$ × Area of actual plot

Area of actual plot = 768 × 20000 × 20000 cm²

= $\dfrac{768 \times 20000 \times 20000}{100000 \times 100000}$ km²

= 30.72 km².

Hence, area of plot = 30.72 km².