Total Number of Triangles Formed With Two Sides – Shortcut Using Triangle Inequality
June 24, 2024 2025-06-24 14:32Total Number of Triangles Formed With Two Sides – Shortcut Using Triangle Inequality
Total Number of Triangles Formed With Two Sides – Shortcut Using Triangle Inequality
Amiya Kumar Total Number of Triangles Formed With Two Sides – Shortcut Using Triangle Inequality
Triangle formation rule: For any three sides a, b, and c to form a triangle, they must satisfy the triangle inequality rule:
Shortcut Concept
If two sides of a triangle are given as a and b where a ≤ b, and the third side c is an integer, then the total number of possible integer values of c such that it forms a valid triangle is given by:
(Where a, b ∈ ℕ and a ≤ b)
Why does the formula work?
From triangle inequality: b − a + 1 ≤ c ≤ a + b − 1
So the count of values = (a + b − 1) − (b − a + 1) + 1 = 2a − 1.
Example
If two sides are 4 and 7, take a = 4 (smaller side). Then, total number of triangles = 2×4 − 1 = 7.
5 Practice Questions
Answer
Smaller side a = 5 → Number = 2×5 − 1 = 9Answer
Smaller side a = 3 → Number = 2×3 − 1 = 5Answer
Smaller side a = 6 → Number = 2×6 − 1 = 11Answer
2a − 1 = 15 ⇒ a = 8 → Answer: 8Answer
Smaller side a = 12 → Number = 2×12 − 1 = 23Conclusion
This shortcut is extremely useful for time-bound competitive exams like CAT, SSC, and other aptitude tests. Just identify the smaller of the two given sides, apply 2a − 1, and you’re done!
Keywords: triangle inequality shortcut, number of triangles formed, triangle formation rule, CAT geometry shortcut, triangle with given sides, triangle side condition.
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