if two towers of height h1 and h2 subtend angles of 60 degree and 30 degree respectively at the midpoint - Maths - Some Applications of Trigonometry

Question

if two towers of height h1 and h2 subtend angles of 60 degree
and 30 degree respectively at the midpoint of the line joining
their feet , then the ratio of h1: h2

Gursheen kaur. · Accepted Answer

Let O be the mid-point of the line joining their feet.

$\Rightarrow B C = \frac{1}{2} O B = \frac{1}{2} O C ............ (i) N o w, I n Δ A O B, \tan 60^{0} = \frac{p e r p}{b a s e} = \frac{A B}{O B} \Rightarrow \tan 60^{0} = \frac{h_{1}}{2 B C} [F r o m (i)] \Rightarrow 2 \sqrt{3} B C = h_{1} ............. (i i) a n d I n Δ D O C, \tan 30^{0} = \frac{p e r p}{b a s e} = \frac{D C}{O C} \Rightarrow \tan 30^{0} = \frac{h_{2}}{2 B C} [F r o m (i)] \Rightarrow \frac{2 B C}{\sqrt{3}} = h_{2} ........... (i i i) h_{1} : h_{2} = 2 \sqrt{3} B C : \frac{2 B C}{\sqrt{3}} \Rightarrow h_{1} : h_{2} = \frac{2 \sqrt{3} B C}{\frac{2 B C}{\sqrt{3}}} = \frac{2 \sqrt{3} B C \times \sqrt{3}}{2 B C} = \frac{\sqrt{3} \times \sqrt{3}}{1} = \frac{3}{1} \Rightarrow h_{1} : h_{2} = 3 : 1$

if two towers of height h1 and h2 subtend angles of 60 degree and 30 degree respectively at the midpoint of the line joining their feet , then the ratio of h1: h2