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

• A. 6
• B. 4
• C. 5
• D. 3

Answer 1

The Correct Option is C

Given three sides of a quadrilateral as 1 cm, 2 cm, and 4 cm, determine the number of possible integer values for the length of the fourth side that allow for the formation of a quadrilateral.

Step 1: Apply Triangle Inequality Theorem

The sum of any two sides of a triangle must be greater than the third side. This principle applies when considering any three sides of a quadrilateral.

The given three sides are: \(1\,\text{cm}, 2\,\text{cm}, 4\,\text{cm}\)

• \(1 + 2 > 4 \Rightarrow 3 > 4\): ❌ False
• \(1 + 4 > 2 \Rightarrow 5 > 2\): ✅ True
• \(2 + 4 > 1 \Rightarrow 6 > 1\): ✅ True

Note: Only two of the three conditions are met. This indicates that a triangle cannot be formed using sides 1, 2, and 4 simultaneously.

Correction: Quadrilateral Inequality

For a quadrilateral, the sum of any three sides must be greater than the fourth side.

Step 2: Determine Possible Ranges for the Fourth Side

Let the fourth side be denoted by \(x\). The quadrilateral inequality states:

\[ \text{Sum of any three sides} > \text{the fourth side} \]

Applying this to the given sides and the fourth side \(x\):

• \(1 + 2 + 4 > x \Rightarrow 7 > x \Rightarrow x < 7\)
• \(x + 1 + 2 > 4 \Rightarrow x > 1\)
• \(x + 1 + 4 > 2 \Rightarrow x > -3\) (This condition is always true for a positive length \(x\))
• \(x + 2 + 4 > 1 \Rightarrow x > -5\) (This condition is always true for a positive length \(x\))

Combining the relevant constraints: \(x > 1\) and \(x < 7\). This gives the range \(1 < x < 7\).

Step 3: List All Integer Values

The integer values of \(x\) that satisfy \(1 < x < 7\) are: \[ 2, 3, 4, 5, 6 \]

Final Answer:

There are 5 possible integer values for the length of the fourth side.

\[ \boxed{5} \]

Correct Option: (C)