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Two vertical poles AL and BM of heights 20 m and 80 m respectively stand apart on a horizontal plane. If A, B be the feet of the poles and AM and BL intersect at P. Find the height of P from the horizontal plane.

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As shown in the figure, AL = 20 m, BH = 80 m. 

Let the height P = PQ = h metres (say) 

Let ∠LBA = ϕ and ∠MAB = θ. Then, 

In ΔMAB = tan θ = \(\frac{MB}{AB}\) = \(\frac{80}{AB}\)           ......(i)

In ΔPQA, = tan θ = \(\frac{PQ}{QA}\) = \(\frac{h}{QA}\)           ......(ii)

∴ From (i) and (ii), \(\frac{80}{AB}\) = \(\frac{h}{QA}\) ⇒ QA = \(\frac{h}{80}\). AB          .....(iii)

In ΔLBA, tan ϕ, \(\frac{LA}{AB}\) = \(\frac{20}{AB}\)                      ......(iv)

In ΔPQB, tan ϕ = \(\frac{PQ}{BQ}\) = \(\frac{h}{BQ}\)                   ......(v)

∴ From (iv) and (v), \(\frac{20}{AB}\) = \(\frac{h}{BQ}\) ⇒ BQ = \(\frac{h}{20}\) AB                 .....(vi)

from (iii) and (vi) BQ + QA = \(\frac{h}{20}\) AB + \(\frac{h}{80}\) AB

⇒ AB = h \(\bigg(\frac{1}{20}+\frac{1}{80}\bigg)\)AB ⇒ 1 = \(\big(\frac{1+4}{80}\big)\)h ⇒ h = \(\frac{80}{5}\) = 16 m.

∴ Height of PQ from horizontal plane = 16 m.

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