# Probability of cutting a rope into three pieces such that the sides form a triangle

Given a rope. We have to find the probability of cutting a rope into 3 pieces such that they form a triangle.

**Answer: **0.25

**Explanation: **

Let the length of rope be 1 unit. We choose two points X and Y on the rope.

**Note: **Formation of triangle is based on **Triangle inequality** i.e. *sum of the lengths of any two sides of a triangle must be greater than the length of the third side*

There are two possibilities of choosing points X and Y on rope:

**Case 1: **X < Y

Length of pieces after choosing points X and Y:

X units, (Y-X) units, (1-Y) units

Below line diagram shows the partition rope.

The 3 possible combination for satisfying Triangle Inequality

1. X + (Y-X) > (1-Y)

=> 2Y > 1

=> Y > (1/2)

2. X + (1-Y) > (Y-X)

=> 2X + 1 > 2Y

=> Y < X + (1/2)

3. (Y-X) + (1-Y) > X

=> 2X < 1

=> X < 1/2

Forming a graph using above 3 conditions and X<Y

The area in the region **1** of the graph is the required area which is: 1/2*1/2*1/2 = 1/8

**Case 2: **X > Y

Length of pieces after choosing points X and Y:

Y units, (X-Y) units, (1-X) units

Below line diagram shows the partition rope.

The 3 possible combinations for satisfying Triangle Inequality

1. Y + (X-Y) > (1-X)

=> 2X > 1

=> X > (1/2)

2. Y + (1-X) > (X-Y)

=> 2Y + 1 > 2X

=> X < Y + (1/2)

3. (X-Y) + (1-X) > Y

=> 2Y < 1

=> Y < 1/2

Forming a Graph using 3 condition and Y<X

The area in the region **5** of the graph is the required area which is: 1/2*1/2*1/2 = 1/8

In both cases we get same required area, thus area required will be: 1/8.

So the Probability of breaking a rope into three pieces such that the sides form a triangle

is given by:

Area required/Total area = (1/8)/(1/2*1*1) = 1/4 = 0.25.

## Recommended Posts:

- Probability that the pieces of a broken stick form a n sided polygon
- Find other two sides of a right angle triangle
- Find other two sides and angles of a right angle triangle
- Find area of triangle if two vectors of two adjacent sides are given
- Check whether right angled triangle is valid or not for large sides
- Find all sides of a right angled triangle from given hypotenuse and area | Set 1
- Find Four points such that they form a square whose sides are parallel to x and y axes
- Possible to form a triangle from array values
- Number of possible pairs of Hypotenuse and Area to form right angled triangle
- Length of rope tied around three equal circles touching each other
- Maximum number of pieces in N cuts
- Count pieces of circle after N cuts
- Biggest Reuleaux Triangle within a Square which is inscribed within a Right angle Triangle
- Biggest Reuleaux Triangle inscribed within a Square inscribed in an equilateral triangle
- Maximize volume of cuboid with given sum of sides

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.