Question

The lengths of all four sides of a quadrilateral are integer values. If three of its sides are of length 1 cm, 2 cm, and 4 cm, then what is the total number of possible lengths of the fourth side?

• A. 5
• B. 6
• C. 3
• D. 4

Answer 1

The Correct Option is A

Step 1: Let the four sides of the quadrilateral be \( a = 1 \), \( b = 2 \), \( c = 4 \), and \( d \) (the length of the fourth side). 
To form a valid quadrilateral, the sum of the lengths of any three sides must be greater than the fourth side (Triangle Inequality Theorem generalized for quadrilaterals). 
This gives us the following inequalities:
\[ a + b + c > d \quad (1) \]
\[ a + b + d > c \quad (2) \]
\[ a + c + d > b \quad (3) \]
\[ b + c + d > a \quad (4) \]
Substituting the known values of \( a \), \( b \), and \( c \):
\[ 1 + 2 + 4 > d \quad \Rightarrow \quad 7 > d \quad \Rightarrow \quad d < 7 \quad (from\ inequality\ 1) \]
\[ 1 + 2 + d > 4 \quad \Rightarrow \quad 3 + d > 4 \quad \Rightarrow \quad d > 1 \quad (from\ inequality\ 2) \]
\[ 1 + 4 + d > 2 \quad \Rightarrow \quad 5 + d > 2 \quad \Rightarrow \quad d > -3 \quad (this\ inequality\ is\ always\ true,\ so\ it\ does\ not\ restrict\ d) \]
\[ 2 + 4 + d > 1 \quad \Rightarrow \quad 6 + d > 1 \quad \Rightarrow \quad d > -5 \quad (this\ inequality\ is\ also\ always\ true) \]
Step 2: Combining the valid inequalities from steps 1 and 2:
\[ 1 < d < 7 \]
Thus, \( d \) can be one of the following integer values: \( 2, 3, 4, 5, 6 \).

For a quadrilateral, use the triangle inequality for all three sets of sides to find the possible values for the fourth side.