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CAT 2022 Slot 3 — QA Question 11

Basics of TrianglesEasy

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

Answer & solution

  • A

    6

  • B

    8

  • 12

  • D

    4

Solution

Easy

Start with an equilateral triangle (all sides 2424). When the slower ship moves 88 km toward the port, the triangle becomes right-angled. Use the 60∘60^\circ angle at the port to locate the right angle, get the speed ratio, then scale up to when the faster ship arrives.

A (port) B B' C
1

Set up. Initially AB=AC=24AB=AC=24 with ∠BAC=60∘\angle BAC=60^\circ (equilateral). The slower ship moves 88 km along its route toward AA, so the new triangle keeps ∠A=60∘\angle A=60^\circ but is now right-angled.

2

Locate the right angle. With ∠A=60∘\angle A=60^\circ fixed, a right angle must sit at the slower ship's new position. In a right triangle with a 60∘60^\circ and a 90∘90^\circ angle, the side opposite 30∘30^\circ is half the hypotenuse:

AC=12⋅AB=12⋅24=8 km (slower ship’s distance left)AC=\tfrac12\cdot AB=\tfrac12\cdot24=8\text{ km (slower ship's distance left)}
3

Find the faster ship's travel FF. Step 2 says the faster ship's remaining distance to the port is AC=8AC=8 km. Since it started 2424 km away, it has already travelled:

F=24−8=16 km in the same elapsed timeF=24-8=16\text{ km in the same elapsed time}
4

Speed ratio (same time elapsed): faster : slower =16:8=2:1=16:8=2:1.

When the faster ship covers its full 2424 km to port, the slower one (half the speed) covers 1212 km, leaving:

24−12=12 km from the port24-12=12\text{ km from the port}
Distance of the other ship from port=12 km\text{Distance of the other ship from port}=\mathbf{12}\text{ km}
CAT 2022 Slot 3 QA Q11: Two ships are approaching a port along straight routes at constant speeds. Initially, the two ships and the po — Solution | TheCATExam