text: 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?
  
  Solution
  
  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.
  
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explanation: The document contains a math question and its corresponding solution. The question involves two lighthouses and the calculation of the length of a perpendicular segment from the intersection of two line segments to the ground. The solution uses similar triangles and the crossed ladder theorem to find the length of the segment.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
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  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 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?
  
  Solution
  
  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
  
  Create a FREE account and get:
  
  • All Quant Formulas and shortcuts PDF
  • XAT previous papers with solutions PDF
  • XAT Trial Classes for FREE
  
  ## CAT Verbal Ability Questions | VARC Questions For CAT
  
  Boost your Prep!
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its corresponding solution. The question involves two lighthouses and the calculation of the length of a perpendicular segment from the intersection of two line segments to the ground. The solution uses similar triangles and the crossed ladder theorem to find the length of the segment.
  
  Label: question-and-answer