Geometry Questions for CAT XAT & MBA

Condition for Missing Side in Polygon

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📐 Condition for Missing Side in Polygon Formation

To determine whether a group of sides can form a valid polygon, especially when one side is missing, we use a key geometric principle: the sum of the lengths of any \( n-1 \) sides must be greater than the remaining side.

🔺 1. Triangle (3 sides)

Let two sides be \( a \leq b \). The third side \( c \) must satisfy:

\( b - a < c < b + a \)

✅ This is the basic triangle inequality. The difference of two known sides < missing side < sum of the two known sides.

🔷 2. Quadrilateral (4 sides)

Let known sides be \( a \leq b \leq c \). For the unknown fourth side \( d \):

\( c - (a + b) < d < c + (a + b) \)

✅ The sum of any three sides must be greater than the fourth. Here, the range of \( d \) is derived from that logic.

🔷 3. Pentagon (5 sides)

Let known sides be \( a \leq b \leq c \leq d \). For the unknown fifth side \( e \):

\( d - (a + b + c) < e < d + (a + b + c) \)

✅ The inequality ensures polygon closure: the sum of any 4 sides must be greater than the fifth.

📌 General Rule for Any Polygon with a Missing Side

If \( n-1 \) sides are known as \( a_1 \leq a_2 \leq \dots \leq a_{n-1} \), then the missing \( n^\text{th} \) side \( x \) must satisfy:

\( a_{n-1} - (a_1 + a_2 + \dots + a_{n-2}) < x < a_{n-1} + (a_1 + a_2 + \dots + a_{n-2}) \)

💡 Always sort the known sides in increasing order before applying the condition. These rules are useful in solving problems in geometry, CAT, SSC, Olympiads, and more.

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