# The angles of depression of the top and bottom of a tower as seen from the top of a 60√3-m-high cliff are 45° and 60° respectively. Find the height of the tower.

Let AB be the cliff and CD be the tower. Join B and C. Draw a line AF parallel to BC. Now, given that the angles of depression of the top and bottom of the tower from the top of the cliff are 45° and 60° respectively. Hence, FAD = 45° and FAC = 60°. Draw a line DE from CD to AB parallel to BC. Also, join A, D and A, C. We get two right-angled triangles ADE and ABC with right angles at E and B respectively. Also, ADE = FAD = 45°, and ACB = FAC = 60°. We are also given that the height of the cliff AB is 60√3 m. We are to find the height of the tower, that is, CD.

We first find the value of BC from the ∆ABC, using the trigonometric ratio tan.

In ∆ABC,

or,

Clearly, ED = BC. Then, we will find the value of AE from ∆ADE using the trigonometric ratio tan. In ∆ADE,

or,

The height of the tower = DC = 60√3m-60m = 43.92 m.

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