Geometry

Triangles

Hard

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

$12$

$4$

$6$

$8$

✅ Correct Option: 1

We need to create a clear, step-by-step solution that explains the geometry and reasoning behind each step.

Let's start by visualizing what's happening:

Initially two ships and a port form an equilateral triangle with all sides = 24 km

Both ships are moving toward the port along straight paths at constant speeds

We need to find which ship is faster and determine their speed ratio

Let's call the ships Ship A (slower) and Ship B (faster).

After Ship A moves 8 km:

Ship A is now 16 km from the port (24 - 8 = 16)

Ship B has moved some distance and is now 24 - d km from the port

The new triangle formed is right-angled

Here's the key insight: When the triangle becomes right-angled, we need to determine where the right angle is located.

Since Ship A moved less distance (it's slower), the right angle must be at Ship B's new position.

Using the 30-60-90 triangle property:

In our right-angled triangle, if the right angle is at Ship B's position

One leg = distance from port to Ship B = 8 km

Hypotenuse = distance from port to Ship A = 16 km

This gives us a 30-60-90 triangle (since 8:16 = 1:2)

Therefore: Ship B has moved 24 - 8 = 16 km when Ship A moved 8 km.

Speed ratio calculation:

Ship A (slower) moved: 8 km

Ship B (faster) moved: 16 km

Speed ratio = 8:16 = 1:2

This means Ship B is twice as fast as Ship A.

Current positions after the right-angled triangle formation:

Ship A: 16 km from port

Ship B: 8 km from port

Time for Ship B to reach the port:

Ship B needs to cover remaining 8 km

In this same time, Ship A will cover: 8 ÷ 2 = 4 km (since Ship A is half as fast)

Final position of Ship A:

Current distance: 16 km

Distance covered: 4 km

Remaining distance = 16 - 4 = 12 km

When the faster ship reaches the port, the distance between the other ship and the port will be 12 km.

Key Learning: This problem combines geometry (equilateral and right triangles) with relative motion. The crucial step was recognizing that the 30-60-90 triangle relationship helped us determine the exact positions after the triangle became right-angled.

CAT 2024 Slot 1

ABCD is a rectangle with sides AB = 56 cm and BC = 45 cm, and E is the midpoint of side CD. Then, the length, in cm, of radius of in circle of $\triangle$ ADE is

CAT 2023 Slot 1

In a right-angled triangle $\triangle A B C$ , the altitude $A B$ is $5 \mathrm{~cm}$ , and the base $B C$ is $12 \mathrm{~cm}$ . $P$ and $Q$ are two points on $B C$ such that the areas of $\triangle \mathrm{ABP}, \triangle \mathrm{ABQ}$ and $\triangle \mathrm{ABC}$ are in arithmetic progression. If the area of $\triangle \mathrm{ABC}$ is $1.5$ times the area of $\triangle \mathrm{ABP}$ , the length of PQ in cm , is

CAT 2018 Slot 2

A triangle $ABC$ has area $32$ sq units and its side $BC$ , of length $8$ units, lies on the line $\mathrm{x}=4$ . Then the shortest possible distance between $A$ and the point $(0,0)$ is

CAT 2022 Slot 2

The length of each side of an equilateral triangle ABC is $3 \mathrm{~cm}$ . Let D be a point on BC such that the area of triangle ADC is half the area of triangle $A B D$ . Then the length of $A D$ , in cm , is

CAT 2021 Slot 2

Let $D$ and $E$ be points on sides $A B$ and $A C$ , respectively, of a triangle $A B C$ , such that $A D: B D=2: 1$ and $A E: C E=2: 3$ . If the area of the triangle $A D E$ is $8 \mathrm{ sq } \mathrm{cm}$ , then the area of the triangle $ABC$ , in sq cm , is

CAT 2022 Slot 2

In triangle ABC, altitudes AD and BE are drawn to the corresponding bases. If $\angle BAC = 45^\circ$ and $\angle ABC = \theta$ , then $\frac{AD}{BE}$ equals

Solution

Related questions:

Keyboard Shortcuts

CAT Past Year Questions (Topic-Wise)