Question:medium

Two ships are approaching a port along straight routes at constant speeds. Initially,the two ships and the port formed an equilateral triangle with sides of length 24 km. When the slower ship travelled 8 km,the triangle formed by the new positions of the two ships and the port became right-angled. When the faster ship reaches the port, the distance,in km,between the other ship and the port will be

Updated On: Jan 15, 2026
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The Correct Option is C

Solution and Explanation

Initially, two ships and a port form an equilateral triangle with a side length of 24 km.

Defined Points:

  • \( A \): Port, initial position \( (0, 0) \)
  • \( B \): Slower ship, initial position \( (24, 0) \)
  • \( C \): Faster ship, initial position at the top vertex of the triangle

Given triangle ABC is equilateral with side 24 km, the coordinates of \( C \) are: \[ C = \left(12\sqrt{3}, 12\right) \] (Triangle positioned symmetrically relative to base AB.)

Movement and Transformation:

  • The slower ship moves 8 km towards port \( A \), resulting in a new position \( P = (16, 0) \)
  • The faster ship moves such that the triangle formed by \( A, P, Q \) is a right-angled triangle at \( P \)

Pythagorean Theorem Application:

Since triangle \( APQ \) is right-angled at \( P \) and \( AP = 16 \), the following must hold: \[ PQ = AQ = 16 \]

Speed Ratio Analysis:

The slower ship travels 8 km while the faster ship travels 24 km, indicating a speed ratio of \( 1:3 \). This implies both ships commence movement simultaneously, and the faster ship reaches the port precisely when the slower ship has covered 8 km.

Final Geometric Configuration:

At this point, triangle \( APQ \) is right-angled at \( P \), and \( AQ = 16 \). Therefore: \[ \text{Distance from Q to port A} = AQ = 16 \text{ km} \] As the triangle is right-angled at \( P \), the side \( PQ \) is perpendicular. Using triangle properties: \[ PQ = 12 \text{ km} \] This is the perpendicular distance from Q to the base AP.

✅ Final Answer: The distance between the remaining ship and the port is 12 km.

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