On a map drawn to a scale of 1 : 50000, a rectangular plot of land ABCD has the following dimensions. AB = 6 cm; BC = 8 cm. Find :

(i) the actual length of the diagonal AC of the plot in km.

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

Since map is drawn to a scale of 1 : 50000.

∴ k (Scale factor) = 50000.

Length of the diagonal AC of the rectangle can be given by pythagoras theorem i.e. $\sqrt{AB^2 + BC^2}$.

Putting values we get,

$\Rightarrow AC = \sqrt{AB^2 + BC^2} \\[1em] \Rightarrow AC = \sqrt{6^2 + 8^2} \\[1em] \Rightarrow AC = \sqrt{36 + 64} \\[1em] \Rightarrow AC = \sqrt{100} \\[1em] \Rightarrow AC = 10 \text{ cm}.$

Actual length of diagonal = k × length of diagonal in model.

$= 50000 \times 10 \\[1em] = 500000 \text{ cm} \\[1em] = 500000 \times 10^{-5} \text{ km} \\[1em] = 5 \text{ km}$

=50000×10=500000 cm=500000×10−5 km=5 km

Hence, actual length of diagonal = 5 km.

(ii) Area of the model ABCD = AB × BC = 6 × 8 = 48 cm².

Area of the actual plot = k² × (Area of the model)
= (50000)² x 48
= 25 x 10⁸ x 48
= 1200 x 10⁸
= 12 x 10¹⁰ cm²

We know that 1 cm² = 10^-10 km².

∴ Actual area of plot = 12 × 10¹⁰ × 10^-10 km² = 12 km².

Hence, the actual area of the plot is 12 km².