Question

On a map, drawn to a scale of 1:25000, a triangular plot LMN of land has the following measurements:
$LM=6cm,MN=8cmand\angle LMN={90}^0.$ Calculate:
(i) the actual lengths of MN and LN in kilometers,
(ii) the actual area of the plot in sq. sm.

Answer 1

We have,

The map is drawn to the scale of $1:25000$

$\therefore k=\frac{1}{25000}$

(1). As shown in the map

We have,

$LM=6cm.$

$MN=8cm.$

We know that,

By Pythagoras theorem,

$L{N}^2=L{M}^2+M{N}^2$

$=6^2+8^2$

$=36+64$

$100$

$L{N}^2={10}^2$

$LN=10cm.$

Now length of diagonal of the map $=k\times$ (length of diagonal of Plot)

$\Rightarrow LN=\frac{1}{25000}\times$ (length of diagonal of Plot)

$\Rightarrow 10=\frac{1}{25000}\times$ (length of diagonal of Plot)

length of diagonal of Plot

$=25000\times 10=250000$

$=\frac{250000}{100\times 1000}$

$=2.5km.$

(2).We have,

Area of the plot on the map$=6\times 8=48c{m}^2$

Actual area of the map$=k^2\times$ Area of the plot on the plot

$48={\left(\frac{1}{25000}\right)}^2\times$ Area of the plot on the plot

Area of the plot on the plot

$=\frac{25000\times 25000\times 48}{100\times 100}{m}^2$

$=\frac{250\times 250\times 48}{1000\times 1000}k{m}^2$

$=3k{m}^2$

Hence, this is the answer.