Question

Two houses are collinear with the base of a tower and are at distances 3 m and 12 m (on the same side) from the base of the tower. The angles of elevation from these two houses of the top of the tower are complementary. What is the height of the tower?

(a) 4 m

(b) 6 m

(c) 7.5 m

(d) 36 m

Answer 1

Correct option is (b) 6 m

Let the height of tower = h

Angle of elevation at 3 m distance = α

Angle of elevation at 12 m distance = 90−α

Now, The line of sight, the ground and the tower will form a right angled triangle.

Thus, tan angle of elevation = \(\frac {\text{height }}{\text{distance}}\) 

Thus, considering the 3 m distance,

tan α = \(\frac h3\)    .....(1)

Considering the 12 m distance,

tan(90 − α) = \(\frac h{12}\)    ......(2)

Multiplying the two equations,

tan α tan(90 − α) = \(\frac h3.\frac h{12}\)

tan α cot α = \(\frac h3.\frac h{12}\)

1 = \(\frac{h^2}{36}\)

h = 6 m