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Check if four segments form a rectangle

Last Updated : 08 Feb, 2023
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We are given four segments as a pair of coordinates of their end points. We need to tell whether those four line segments make a rectangle or not. 
Examples: 
 

Input : segments[] =  [(4, 2), (7, 5),
                       (2, 4), (4, 2),
                       (2, 4), (5, 7),
                       (5, 7), (7, 5)]
Output : Yes
Given these segment make a rectangle of length 3X2.

Input : segment[] = [(7, 0), (10, 0),
                     (7, 0), (7, 3),
                     (7, 3), (10, 2),
                     (10, 2), (10, 0)]
Output : Not
These segments do not make a rectangle.

Above examples are shown in below diagram.


This problem is mainly an extension of How to check if given four points form a square
 


We can solve this problem by using properties of a rectangle. First, we check total unique end points of segments, if count of these points is not equal to 4 then the line segment can’t make a rectangle. Then we check distances between all pair of points, there should be at most 3 different distances, one for diagonal and two for sides and at the end we will check the relation among these three distances, for line segments to make a rectangle these distance should satisfy Pythagorean relation because sides and diagonal of rectangle makes a right angle triangle. If they satisfy mentioned conditions then we will flag polygon made by line segment as rectangle otherwise not.
 

CPP 
// C++ program to check whether it is possible 
// to make a rectangle from 4 segments 
#include <bits/stdc++.h> 
using namespace std; 
#define N 4 

// structure to represent a segment 
struct Segment 
{ 
    int ax, ay; 
    int bx, by; 
}; 

// Utility method to return square of distance 
// between two points 
int getDis(pair<int, int> a, pair<int, int> b) 
{ 
    return (a.first - b.first)*(a.first - b.first) + 
        (a.second - b.second)*(a.second - b.second); 
} 

// method returns true if line Segments make 
// a rectangle 
bool isPossibleRectangle(Segment segments[]) 
{ 
    set< pair<int, int> > st; 

    // putting all end points in a set to 
    // count total unique points 
    for (int i = 0; i < N; i++) 
    { 
        st.insert(make_pair(segments[i].ax, segments[i].ay)); 
        st.insert(make_pair(segments[i].bx, segments[i].by)); 
    } 

    // If total unique points are not 4, then 
    // they can't make a rectangle 
    if (st.size() != 4) 
        return false; 

    // dist will store unique 'square of distances' 
    set<int> dist; 

    // calculating distance between all pair of 
    // end points of line segments 
    for (auto it1=st.begin(); it1!=st.end(); it1++) 
        for (auto it2=st.begin(); it2!=st.end(); it2++) 
            if (*it1 != *it2) 
                dist.insert(getDis(*it1, *it2)); 

    // if total unique distance are more than 3, 
    // then line segment can't make a rectangle 
    if (dist.size() > 3) 
        return false; 

    // copying distance into array. Note that set maintains 
    // sorted order. 
    int distance[3]; 
    int i = 0; 
    for (auto it = dist.begin(); it != dist.end(); it++) 
        distance[i++] = *it; 

    // If line seqments form a square 
    if (dist.size() == 2) 
    return (2*distance[0] == distance[1]); 

    // distance of sides should satisfy pythagorean 
    // theorem 
    return (distance[0] + distance[1] == distance[2]); 
} 

// Driver code to test above methods 
int main() 
{ 
    Segment segments[] = 
    { 
        {4, 2, 7, 5}, 
        {2, 4, 4, 2}, 
        {2, 4, 5, 7}, 
        {5, 7, 7, 5} 
    }; 

    (isPossibleRectangle(segments))?cout << "Yes\n":cout << "No\n"; 
}
Java 
/*package whatever //do not write package name here */
import java.io.*;
import java.util.*;

// java program to check whether it is possible 
// to make a rectangle from 4 segments 
public class Main {
  static int N = 4;

  // Utility method to return square of distance 
  // between two points 
  public static int getDis(String a, String b) 
  { 
    String[] a1 = a.split(",");
    String[] b1 = b.split(",");
    return (Integer.parseInt(a1[0]) - Integer.parseInt(b1[0]))*(Integer.parseInt(a1[0]) - Integer.parseInt(b1[0])) + (Integer.parseInt(a1[1]) - Integer.parseInt(b1[1]))*(Integer.parseInt(a1[1]) - Integer.parseInt(b1[1]));
  } 

  // method returns true if line Segments make 
  // a rectangle 
  public static boolean isPossibleRectangle(int[][] segments) 
  { 
    HashSet<String> st = new HashSet<>();

    // putting all end points in a set to 
    // count total unique points 
    for (int i = 0; i < N; i++) 
    { 
      st.add(segments[i][0] + "," + segments[i][1]); 
      st.add(segments[i][2] + "," + segments[i][3]); 
    } 

    // If total unique points are not 4, then 
    // they can't make a rectangle 
    if (st.size() != 4) 
      return false; 

    // dist will store unique 'square of distances' 
    HashSet<Integer> dist = new HashSet<>();

    // calculating distance between all pair of 
    // end points of line segments 
    for(String it1 : st){
      for(String it2 : st){
        if(it1 != it2){
          dist.add(getDis(it1, it2));
        }
      }
    }

    // if total unique distance are more than 3, 
    // then line segment can't make a rectangle 
    if (dist.size() > 3) 
      return false; 

    // copying distance into array. Note that set maintains 
    // sorted order. 
    int[] distance = new int[3]; 
    int i = 0; 
    for(int it: dist){ 
      distance[i] = it; 
      i = i + 1;
    }

    // If line seqments form a square 
    if (dist.size() == 2) 
      return (2*distance[0] == distance[1]); 

    // distance of sides should satisfy pythagorean 
    // theorem 
    return (distance[0] + distance[1] == distance[2]); 
  } 


  public static void main(String[] args) {
    int[][] segments = 
    { 
      {4, 2, 7, 5}, 
      {2, 4, 4, 2}, 
      {2, 4, 5, 7}, 
      {5, 7, 7, 5} 
    }; 

    if(isPossibleRectangle(segments) == true){
      System.out.println("Yes");
    }
    else{
      System.out.println("No");
    }
  }
}

// The code is contributed by Nidhi goel.
Python3 
# Python program to check whether it is possible 
# to make a rectangle from 4 segments 
N = 4

# Utility method to return square of distance 
# between two points 
def getDis(a, b): 
    return (a[0] - b[0])*(a[0] - b[0]) + (a[1] - b[1])*(a[1] - b[1])

# method returns true if line Segments make 
# a rectangle 
def isPossibleRectangle(segments): 

    st = set()

    # putting all end points in a set to 
    # count total unique points 
    for i in range(N): 
        st.add((segments[i][0], segments[i][1]))
        st.add((segments[i][2], segments[i][3])) 

        
    # If total unique points are not 4, then 
    # they can't make a rectangle 
    if len(st) != 4:
        return False

    # dist will store unique 'square of distances' 
    dist = set()

    # calculating distance between all pair of 
    # end points of line segments 
    for it1 in st:
        for it2 in st:
            if it1 != it2:
                dist.add(getDis(it1, it2))

    # if total unique distance are more than 3, 
    # then line segment can't make a rectangle 
    if len(dist) > 3:
        return False

    # copying distance into array. Note that set maintains 
    # sorted order. 
    distance = []
    for x in dist:
        distance.append(x)
    
    # Sort the distance list, as set in python, does not sort the elements by default. 
    distance.sort()
    
    # If line seqments form a square 
    if len(dist) ==2 :
        return (2*distance[0] == distance[1])

    # distance of sides should satisfy pythagorean 
    # theorem 
    return (distance[0] + distance[1] == distance[2])

# Driver code to test above methods 
segments = [
        [4, 2, 7, 5], 
        [2, 4, 4, 2], 
        [2, 4, 5, 7], 
        [5, 7, 7, 5] 
]

if(isPossibleRectangle(segments) == True): 
    print("Yes")
else:
    print("No") 
    
# The code is contributed by Nidhi goel.
C# 
// C# program to check whether it is possible 
// to make a rectangle from 4 segments 
using System;
using System.Collections.Generic;

class GFG {

  public static int N = 4;

  // Utility method to return square of distance 
  // between two points 
  public static int getDis(KeyValuePair<int,int> a, KeyValuePair<int,int> b) 
  { 
    return (a.Key - b.Key)*(a.Key - b.Key) + 
      (a.Value - b.Value)*(a.Value - b.Value); 
  } 

  // method returns true if line Segments make 
  // a rectangle 
  public static bool isPossibleRectangle(int[,] segments) 
  { 
    HashSet<KeyValuePair<int, int>> st = new HashSet<KeyValuePair<int, int>>();

    // putting all end points in a set to 
    // count total unique points 
    for (int j = 0; j < N; j++) 
    { 
      st.Add(new KeyValuePair<int, int>(segments[j,0], segments[j,1]));
      st.Add(new KeyValuePair<int, int>(segments[j,2], segments[j,3]));
    } 

    // If total unique points are not 4, then 
    // they can't make a rectangle 
    if (st.Count != 4) 
      return false; 

    // dist will store unique 'square of distances' 
    HashSet<int> dist = new HashSet<int>();


    // calculating distance between all pair of 
    // end points of line segments 
    foreach(var it1 in st){
      foreach(var it2 in st){
        if(it1.Key != it2.Key && it1.Value != it2.Value){
          dist.Add(getDis(it1, it2));
        }
      }
    }

    // if total unique distance are more than 3, 
    // then line segment can't make a rectangle 
    if (dist.Count > 3) 
      return false; 

    // copying distance into array. Note that set maintains 
    // sorted order. 
    int[] distance = new int[3]; 
    int i = 0; 
    foreach(var it in dist){
      distance[i] = it;
      i = i + 1;
    } 

    // If line seqments form a square 
    if (dist.Count == 2) 
      return (2*distance[0] == distance[1]); 

    // distance of sides should satisfy pythagorean 
    // theorem 
    return (distance[0] + distance[1] == distance[2]); 
  } 


  // Driver code
  public static void Main()
  {

    int[,] segments = { 
      {4, 2, 7, 5}, 
      {2, 4, 4, 2}, 
      {2, 4, 5, 7}, 
      {5, 7, 7, 5} 
    }; 

    if(isPossibleRectangle(segments) == true){
      Console.WriteLine("Yes");
    }
    else{
      Console.WriteLine("No");
    }
  }
}

// The code is contributed by Nidhi goel.
JavaScript 
// JavaScript program to check whether it is possible 
// to make a rectangle from 4 segments 

const N = 4;

// Utility method to return square of distance 
// between two points 
function getDis(a, b) 
{ 
    return (parseInt(a[0]) - parseInt(b[0]))*(parseInt(a[0]) - parseInt(b[0])) + (parseInt(a[1]) - parseInt(b[1]))*(parseInt(a[1]) - parseInt(b[1])); 
} 

// method returns true if line Segments make 
// a rectangle 
function isPossibleRectangle(segments) 
{   
    let st = new Set();

    // putting all end points in a set to 
    // count total unique points 
    for (let i = 0; i < N; i++) 
    { 
        let tmp1 = [segments[i][0], segments[i][1]];
        let tmp2 = [segments[i][2], segments[i][3]];
        st.add(tmp1.join('')); 
        st.add(tmp2.join('')); 
    }

    // If total unique points are not 4, then 
    // they can't make a rectangle 
    if (st.size != 4)
    {
        return false; 
    }
        
    // dist will store unique 'square of distances' 
    let dist = new Set();

    // calculating distance between all pair of 
    // end points of line segments 
    for(let it1 of st)
    {
        for(let it2 of st)
        {
            if(it1 !== it2)
            {
                dist.add(getDis(it1.split(''), it2.split('')));
            }
        }
    }

    // if total unique distance are more than 3, 
    // then line segment can't make a rectangle 
    if (dist.size > 3)
    {
        return false; 
    } 
        
    // copying distance into array. Note that set maintains 
    // sorted order. 
    let distance = new Array();
    for (let x of dist)
    {
        distance.push(x);
    }
        
    // If line seqments form a square 
    if (dist.size === 2)
    {
        return (2*distance[0] == distance[1]); 
    }

    // distance of sides should satisfy pythagorean 
    // theorem 
    return (distance[0] + distance[1] == distance[2]); 
} 

// Driver code to test above methods 
{ 
    let segments = [
        [4, 2, 7, 5], 
        [2, 4, 4, 2], 
        [2, 4, 5, 7], 
        [5, 7, 7, 5] ]

    if(isPossibleRectangle(segments)){
        console.log("Yes");
    }
    else{
        console.log("No");
    }
} 

// The code is contributed by  Nidhi Goel

Output: 
 

Yes

Time Complexity: O(n2logn) 
Auxiliary Space: O(n)


 


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Minimum Perimeter of n blocks

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