Question 51

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?


Triangle ACC' is similar to triangle ABB'

Considering AC = a, BC = b, CC' = h, AA' is given as 60, BB' is given to be 40.

AC/AB = CC'/BB' = h/40.

$$\left(\frac{a}{a+b}\right)\ =\ \frac{h}{40}$$  (1)

Similarly triangle BCC' is similar to BAA'.

BC/AB = CC'/AA' = h/60.

$$\left(\frac{b}{a+b}\right)\ =\ \frac{h}{60}$$  (2)

Adding (1) and (2).

$$\frac{h}{40}+\frac{h}{60\ }=\ 1$$

1/h = $$\left(\frac{1}{40}+\frac{1}{60}\right)$$

h = 24

Using crossed ladder theorem 

$$\frac{1}{CC'}=\frac{1}{AA'}+\frac{1}{BB'}$$=1/60 +1/40 =5/120 =24.  

Video Solution


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