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Question:

Two lighthouses, located at points A and B on the earth, are 60 feet and 40 feet tall respectively. Each lighthouse is perfectly vertical and the land connecting A and B is perfectly flat. The topmost point of the lighthouse at A is A’ and of the lighthouse at B is B’. Draw line segments A’B and B’A, and let them intersect at point C’. Drop a perpendicular from C’ to touch the earth at point C. What is the length of CC’ in feet?

Explanation:

∆BAA’ ~ ∆BCC’
∴ $\frac{B C}{B A} = \frac{C C'}{A A'}$ …(1)

Also, ∆ABB’ ~ ∆ACC’
∴ $\frac{A C}{A B} = \frac{C C'}{B B'}$ …(2)

Adding (1) and (2) we get

$\frac{A C + B C}{A B} = \frac{C C'}{A A'} + \frac{C C'}{B B'}$

⇒ 1 = $\frac{C C'}{A A'} + \frac{C C'}{B B'}$

⇒ $\frac{1}{C C'} = \frac{1}{A A'} + \frac{1}{B B'}$

⇒ $\frac{1}{C C'} = \frac{1}{60'} + \frac{1}{40}$ = $\frac{1}{24}$

∴ CC’ = 24.

Hence, option (c).

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