Question

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

Answer 1

$\left(i\right)$ Using Trigonometric ratios :

In $△CDE$,

$\tan \theta =\frac{DE}{DC}$

$\tan \theta =\frac{1.6}{4.8}=\frac{1}{3}$ ----- ( 1 )

In $△CBA$,

$\tan \theta =\frac{AB}{BC}$

$\Rightarrow$ $\frac{1}{3}=\frac{AB}{3.2+4.8}$ [ From ( 1 ) ]

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

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

$\therefore$ The height of the pole is $2.67m$

$\left(b\right)$ Using similar triangles.

In $△CDE$ and $△CBA$

$\angle CDE=\angle CBA={90}^o$

$\angle DCE=\angle BCA$ [ Common angle ]

$\therefore$ $△CDE\sim △CBA$ [ By AA similarity criteria ]

$\therefore$ $\frac{DE}{AB}=\frac{CD}{BC}$

$\Rightarrow$ $\frac{1.6}{AB}=\frac{4.8}{8}$

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

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