PQR is a triangular park with PQ = PR = 200 m.

Question

PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles
of elevation of the top of the tower at P, Q and R are respectively 45$$^\circ $$, 30$$^\circ $$ and 30$$^\circ $$, then the height of the tower (in m) is :

Accepted Answer

Let height of tower $TM = h$

From triangle $PMT,$

$$\tan {45^o} = {h \over {PM}}$$

$$ \Rightarrow \,\,\,PM = h$$

From triangle $TRM,$

$$\tan {30^o} = {{TM} \over {RM}}$$

$$ \Rightarrow \,\,\,RM = \sqrt3 h$$

From triangle. $PMR,$

$$P{M^2} + M{R^2} = {\left( {200} \right)^2}$$

$$ \Rightarrow \,\,\,h{}^2 + {\left( {\sqrt 3 h} \right)^2} = {200^2}$$

$$ \Rightarrow \,\,\,4{h^2} = {200^2}$$

$$ \Rightarrow \,\,\,{h^2} = {{{{200}^2}} \over 4}$$

$$ \Rightarrow \,\,\,h = {{200} \over 2} = 100\,m$$

$$\therefore\,\,\,$$ height of tower $=100$ $m.$

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