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:
Answer & explanation
Correct answer: option 2
\(\angle\)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.