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 how many integer values are possible for the length of the fourth side such that a quadrilateral can be formed.

Step 1: Apply Triangle Inequality Theorem

For any three sides of a triangle, the sum of any two sides must be greater than the third. This condition also applies when selecting three sides from a quadrilateral.

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

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

Oops! Only two of the three conditions are satisfied. This implies: **a triangle cannot be formed** using sides 1, 2, and 4 simultaneously.

But wait — this needs a correction:

Actually, since it's a quadrilateral, and we are checking for **any triangle** formed using three sides of the quadrilateral, we only need to make sure that when three sides are selected, they satisfy the triangle inequality with the fourth.

Step 2: Determine Possible Ranges for the Fourth Side

Let the fourth side be \(x\). The quadrilateral inequality says: \[ \text{Sum of any three sides} > \text{the fourth side} \] So:

• \(1 + 2 + 4 > x \Rightarrow 7 > x \Rightarrow x < 7\)
• \(x + 1 + 2 > 4 \Rightarrow x > 1\)
• \(x + 1 + 4 > 2 \Rightarrow x > -3\) (Always true)
• \(x + 2 + 4 > 1 \Rightarrow x > -5\) (Always true)

So we combine the valid constraints: \[ x > 1 \quad \text{and} \quad x < 7 \Rightarrow 1 < x < 7 \]

Step 3: List All Integer Values

Integer values of \(x\) satisfying \(1 < x < 7\) are: \[ 2, 3, 4, 5, 6 \]

Final Answer:

There are 5 possible integer values for the fourth side.

\[ \boxed{5} \]

Correct Option: (C)

Answer 2

Given three sides of a quadrilateral as 1 cm, 2 cm, and 4 cm, determine how many possible integer values the fourth side (denoted \( x \)) can have, such that a quadrilateral can be formed.

Key Rule:

In any polygon, the length of the longest side must be **less than** the sum of the remaining sides. For a quadrilateral with side lengths \( a, b, c, x \), we must ensure: \[ \text{Longest side} < \text{Sum of other three sides} \]

Case I: Assume \( x \) is the longest side

Then: \[ x < 1 + 2 + 4 = 7 \] So the possible integer values for \( x \): 4, 5, 6

Case II: Assume 4 is the longest side

We require: \[ 4 < x + 1 + 2 \Rightarrow 4 < x + 3 \Rightarrow x > 1 \] Since \( x \) must also be less than 4 (to keep 4 as the largest), we get: \[ 1 < x < 4 \Rightarrow x = 2, 3 \]

Combine All Possible Values:

From Case I: \( x = 4, 5, 6 \)
From Case II: \( x = 2, 3 \)

Total possible values for \( x \): \[ \boxed{2, 3, 4, 5, 6} \Rightarrow \boxed{5 \text{ values}} \]

Final Answer:

\[ \boxed{5} \]