Question:
The lengths of all four sides of a quadrilateral are integer valued.If three of its sides are of length 1cm,2cm and 4cm,then the total number of possible lengths of the fourth side is
The lengths of all four sides of a quadrilateral are integer valued.If three of its sides are of length 1cm,2cm and 4cm,then the total number of possible lengths of the fourth side is
Updated On: Sep 30, 2024
6
4
5
3
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The Correct Option is C
Approach Solution - 1
The correct answer is C:5
To determine the possible lengths of the fourth side of the quadrilateral,we need to consider the triangle inequality theorem.According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.This principle also holds for quadrilaterals.
Given the lengths of the three sides as 1cm, 2cm, and 4cm, we can consider all the possible combinations of these sides as two sides of the quadrilateral.Let's check each case:
1cm+2cm> 4cm (True)
1cm+4cm> 2cm (True)
2cm+4cm> 1 cm (True)
Since all three combinations satisfy the triangle inequality theorem,any of them can form a valid triangle,and consequently,a quadrilateral with the given side lengths.
Now,let's consider the possible lengths of the fourth side for each combination:
For 1cm+2cm> 4cm:
The fourth side can have a length between |1cm-2cm|+1 and 1cm+2cm-1.
This gives us a range of possible lengths: 1 to 2cm.
For 1cm+4cm> 2cm:
The fourth side can have a length between |1cm - 4cm|+1 and 1cm+ 4cm- 1.
This gives us a range of possible lengths: 2 to 4cm.
For 2cm+ 4cm> 1cm:
The fourth side can have a length between |2cm- 4cm|+ 1 and 2cm+ 4cm- 1.
This gives us a range of possible lengths: 3 to 5cm.
Now,let's combine the ranges from all three cases: 1-2cm,2-4cm, and 3-5cm.
The possible integer lengths for the fourth side are: 1, 2, 3, 4, 5.
Therefore, the total number of possible lengths for the fourth side is 5.
To determine the possible lengths of the fourth side of the quadrilateral,we need to consider the triangle inequality theorem.According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.This principle also holds for quadrilaterals.
Given the lengths of the three sides as 1cm, 2cm, and 4cm, we can consider all the possible combinations of these sides as two sides of the quadrilateral.Let's check each case:
1cm+2cm> 4cm (True)
1cm+4cm> 2cm (True)
2cm+4cm> 1 cm (True)
Since all three combinations satisfy the triangle inequality theorem,any of them can form a valid triangle,and consequently,a quadrilateral with the given side lengths.
Now,let's consider the possible lengths of the fourth side for each combination:
For 1cm+2cm> 4cm:
The fourth side can have a length between |1cm-2cm|+1 and 1cm+2cm-1.
This gives us a range of possible lengths: 1 to 2cm.
For 1cm+4cm> 2cm:
The fourth side can have a length between |1cm - 4cm|+1 and 1cm+ 4cm- 1.
This gives us a range of possible lengths: 2 to 4cm.
For 2cm+ 4cm> 1cm:
The fourth side can have a length between |2cm- 4cm|+ 1 and 2cm+ 4cm- 1.
This gives us a range of possible lengths: 3 to 5cm.
Now,let's combine the ranges from all three cases: 1-2cm,2-4cm, and 3-5cm.
The possible integer lengths for the fourth side are: 1, 2, 3, 4, 5.
Therefore, the total number of possible lengths for the fourth side is 5.
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Approach Solution -2
The longest side length in any polygon should be less than the total of the other side lengths.
Let 'x' be the fourth side.
x is the largest side in Case I.
The following numbers can be used for x < 1 + 2 + 6
x < 7
x: 4, 5, 6
In case ii, x is smaller than 4 and 4 is the greatest side.
4 < x + 1 + 2 x > One x can accept the values. two or three
Thus, x can have the following values: 2, 3, 4, 5, 6.
x has five possible values.
Let 'x' be the fourth side.
x is the largest side in Case I.
The following numbers can be used for x < 1 + 2 + 6
x < 7
x: 4, 5, 6
In case ii, x is smaller than 4 and 4 is the greatest side.
4 < x + 1 + 2 x > One x can accept the values. two or three
Thus, x can have the following values: 2, 3, 4, 5, 6.
x has five possible values.
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