Prove that the height of the tower is 6m.

Question

The angles of elevation of the top of a tower from two points at a distance of 4m and 9m from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is 6m.

Answer 1

Let AB be the tower.
C and D be the two points with distance 4 m and 9 m from the base respectively.
A/q,
In  right ΔABC,
tan x = AB/BC
⇒ tan x = AB/4
⇒ AB = 4 tan x ... (i)
also,
In  right ΔABD,
tan (90°-x) = AB/BD
⇒ cot x = AB/9
⇒ AB = 9 cot x  ... (ii)
Multiplying  eqn (i) and (ii)
AB² = 9 cot x × 4 tan x
⇒ AB² = 36
⇒ AB = ± 6
Height cannot be negative. Therefore, the height of the tower is 6 m. Hence, Proved.