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?

Accepted Answer

D. By the Crossed Ladder Theorem, if two vertical poles of heights $h_1$ and $h_2$ are erected, the height $h$ of the intersection of the lines connecting top to opposite bases is given by: $$\frac{1}{h} = \frac{1}{h_1} + \frac{1}{h_2}$$ Here $h_1 = 60$ and $h_2 = 40$. $$\frac{1}{h} = \frac{1}{60} + \frac{1}{40} = \frac{2+3}{120} = \frac{5}{120} = \frac{1}{24} \implies h = 24	ext{ feet}.$$

Crossed Ladder Theorem - Lighthouses

Explanation

Practise this under exam conditions