Question

On a map drawn to a scale of $1:25000$, a rectangular plot of land, $ABCD$ has the following measurements $AB=12cm$ and $BG=16cm$. Calculate:
(i) the distance of a diagonal of the plot in km.
(ii) the area of the plot in sq. km.

Answer 1

From the question it is given that,

Map drawn to a scale of $1:25000$

$AB=12cm,BG=16cm$

Consider the $△ABC$,

From the Pythagoras theorem,

$A{C}^2=A{B}^2+B{C}^2$

$AC=\sqrt{(A{B}^2+B{C}^2)}$

$=\sqrt{\left(\right(12{)}^2+\left(16{)}^2\right)}$

$=\sqrt{144}+256$

$=\sqrt{400}$

$=20cm$

Then, area of rectangular plot $ABCD=AB\times BC$

$=12\times 16$

$=192c{m}^2$

(i) We have to find the distance of a diagonal of the plot in km.

Let us assume scale factor $K=1:25000$

$K=1/25000$

Then, length of $AB$ of actual plot $=1/k\times$ length of diagonal of rectangular plot

$=\left(1/\right(1/25000\left)\right)\times 3$

$=25000\times 20$

To covert cm into km divide by $100000$

$=(25000\times 20)/(100\times 1000)$

$=5km$

(ii) We have to find the area of the plot in sq. km.

Then, area of actual plot $=1/k^2\times 192c{m}^2$

$={25000}^2\times 192$

To covert cm into km divide by $(100000{)}^2$

$=(25000\times 25000\times 192)/(100000\times 100000)$

$=12k{m}^2$.