Question

In the given figure, $\angle DAB$ = $\angle ABC$ = ${90}^{\circ }$. Also if $\alpha$ + $\phi$ = ${90}^{\circ }$ and α = 2$\phi$. The the ratio of H and h is ___

Answer 1

$\alpha$ + $\phi$ = ${90}^{\circ }$ -------- (I)

$\alpha$ = 2$\phi$ ------- (II)

Solving I and II, we get $\alpha$ = ${60}^{\circ }$ and $\phi$ = ${30}^{\circ }$

Let the length of base AB be x

In $\Delta ABC$,
$\tan$${60}^{\circ }$ = $\frac{H}{x}=\sqrt{3}$ [$\because$ $\tan$${60}^{\circ }=\sqrt{3}$]

and $\Delta ABD$,
$\tan$${30}^{\circ }$ = $\frac{h}{x}$ [$\because$ $\tan$${30}^{\circ }=\frac{1}{\sqrt{3}}$]

$\Rightarrow$ $\frac{H}{h}$ = 3