The angle of elevation of the top of a tall building from the points M and N at the distances of 72 m and 128 m, respectively, from the base of the building and in the same straight line with it, are complementary. The height of the building (in m) is:

96

\(\angle\)QNP and \(\angle\)QMP are complementary

Let \(\angle\)QMP = \(\theta \)

So, \(\angle\)QNP = (90 - \(\theta \))

Let, the height of the building (PQ) = a m,

In triangle QMP, tan \(\theta \) = \(\frac{PB}{PM}\)

⇒ tan \(\theta \) = \(\frac{a}{72}\)     ..(1.)

In triangle QNP, tan(90 - \(\theta \)) = \(\frac{AB}{AN}\)

⇒ tan(90 - \(\theta \)) = \(\frac{a}{128}\)

⇒ cot \(\theta \) = \(\frac{a}{128}\)

⇒ tan \(\theta \) = \(\frac{128}{a}\)    ..(2.)

So, \(\frac{a}{72}\) = \(\frac{128}{a}\)

⇒ \( {a }^{2 } \) = 128 x 72

⇒ \( {a }^{2 } \) = 9216

⇒ a = \(\sqrt {9216 }\) = 96

Therefore, the height of the building is 96m.