Question

A round balloon of radius r subtends an angle alpha at the eye of the observer while the angle of elevation of it's centre is bita prove that the height of the centre of the balloon is rsinbita×cosecalpha/2

Answer 1

Let the height of centre of the balloon above the ground be h m.

Given, balloon subtends an θ angle at the observes eye. 

∴ ∠ EAD = θ

In ΔACE and ΔACD,

AE = AD  (Length of tangents drawn from an external point to the circle are equal)

AC = AC  (Common) 

CE = CD  (Radius of the circle)

∴ ΔACE  ΔACD  (SSS congruence criterion)

⇒ ∠EAC = ∠DAC  (CPCT)

∴ EAC = DAC = 

In right ΔACD,



In right ΔACB,



Thus, the height of the centre of the balloon is a