Amiya Kumar 
🧮 Triangle Concept – Range of the Third Side
📘 Concept: Range of Third Side in a Triangle
If two sides of a triangle are given as \( a \) and \( b \), then the third side \( c \) must satisfy:
\( |a - b| < c < a + b \)
This inequality comes from the Triangle Inequality Theorem, which states that the sum of any two sides of a triangle must be greater than the third side.
- Lower bound: \( c > |a - b| \)
- Upper bound: \( c < a + b \)
This gives a continuous **range of values** for the third side, which is useful in checking triangle formation and creating number-based MCQs.
🔍 Examples with Images
Example 1: Two sides given: 3 and 10
Then the third side \( c \) satisfies:
\( |10 - 3| < c < 10 + 3 \Rightarrow 7 < c < 13 \)
Example 2: Sides: 10 and 30
\( |30 - 10| < c < 30 + 10 \Rightarrow 20 < c < 40 \)
🧠 Practice Question
Q: Two sides of a triangle are 5 and 10. What is the range of the third side?
\( |10 - 5| < c < 10 + 5 \Rightarrow 5 < c < 15 \)
⚠️ Important Notes
- The third side must be a **positive real number**.
- If you are restricted to natural numbers, count the integers in the open interval (e.g., \( c = 8, 9, 10, 11, 12 \) etc.).
- Always use the triangle inequality rule to check if three given sides can form a triangle.
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