On a map drawn to a scale of 1 : 25000, a triangular plot LMN of land has the following measurements :

LM = 6 cm, MN = 8 cm and ∠LMN = 90°. Calculate :

(i) the actual lengths of MN and LN in kilometers,

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

(i) Since, the model of the triangular plot is made to the scale of 1 : 25000.

k = $\dfrac{1}{25000}$.

Length of MN in model = k × (Actual length of MN)

8 = $\dfrac{1}{25000}$ × Actual length of MN

Actual length of MN = 8 × 25000

= 200000 cm.

= $\dfrac{200000}{100000}$ km.

= 2 km.

Length of LM in model = k × (Actual length of LM)

6 = $\dfrac{1}{25000}$ × Actual length of LM

Actual length of LM = 6 × 25000

= 150000 cm.

= $\dfrac{150000}{100000}$ km.

= 1.5 km.

Using pythagoras theorem:

LN² = LM² + MN²

LN² = (1.5)² + 2²

LN² = 2.25 + 4

LN² = 6.25

LN = $\sqrt{6.25}$

LN = 2.5 km

Hence, MN = 2 km and LN = 2.5 km.

(ii) We know that,

Area of triangle = $\dfrac{1}{2}$ × Base × Height

= $\dfrac{1}{2}$ × LM × MN

= $\dfrac{1}{2}$ × 1.5 × 2

= 1.5 km².

Hence, actual area of the plot = 1.5 sq.km.