Question

A $1.6m$ tall girl stands at a distance of $3.2m$ from a lamp-post and casts a shadow of $4.8m$ on the ground. Find the height of the lamp-post by using (i) trigonometric ratios (ii) property of similar triangles.

Answer 1

$\left(i\right)$ Using trigonometric ratios.

Let $AB$ be the height of lamp post.

Now, in right $△CDE,$

$\Rightarrow \tan \theta =\frac{ED}{DC}=\frac{1.6}{4.8}=\frac{1}{3}⟶\left(1\right)$

In , $△ACB,$

$\Rightarrow \tan \theta =\frac{AB}{BC}=\frac{AB}{3.2+4.8}=\frac{AB}{8}⟶\left(2\right)$

From $\left(1\right)\&\left(2\right)$ we get

$\Rightarrow \frac{1}{3}=\frac{AB}{8}$

$\Rightarrow AB=\frac{8}{3}=2.67m$

$\therefore$ Height of the lamp post $=2.67m$

$\left(ii\right)$ Using similar triangles :

In $△CDE\&△CBA$

$i)\angle CDE=\angle CBA=90°$

$ii)\angle DCE=\angle BCA$ (Common)

$\therefore △CDE\sim △CBA$ ( By AA similarities )

Hence, $\frac{DE}{AB}=\frac{CD}{BC}$

$\Rightarrow AB=\frac{DE\times BC}{CD}$

$=\frac{1.6\times 8}{4.8}$

$=\frac{8}{3}$

$=2.67m$

$\therefore$ Height of lamp post $=2.67m.$

Hence, the answer is $\mathrm{2.67.}$