The scale of a map is 1 : 200000. A plot of land of area 10 km² is to be represented on the map. Find :

(i) the length in km on the ground, represented by 1 cm on the map,

(ii) the area in km² that can be represented by 3 cm² on the map,

(iii) the area on the map representing the plot of land.

(i) Since, the scale of map is 1 : 200000.

1 cm on map = 200000 cm on ground

200000 cm = $\dfrac{200000}{100000}$ = 2km

Hence, 1 cm on map represents 2 km on ground.

(ii) Given,

Area of plot on map = 3 cm²

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

3 = $\Big(\dfrac{1}{200000}\Big)^2$ × Area of actual plot

Area of actual plot = 3 × 200000 × 200000 cm²

= $\dfrac{3 \times 200000 \times 200000}{100000 \times 100000}$ km²

= 12 km^2.

Hence, required area = 12 km².

(iii) Area of plot on map = k² × Area of actual plot

Area of plot on map = $\Big(\dfrac{1}{200000}\Big)^2 \times 10$ km²

Area of plot on map = $\dfrac{10 \times 100000 \times 100000}{200000 \times 200000}$ cm²

= 2.5 cm².

Hence, area on the map representing the plot of land is 2.5 cm².