Amiya Kumar 
Total Number of Isosceles Triangles When Two Sides Are Given
In this post, we’ll explore a fascinating case of triangle formation — how many isosceles triangles can be formed when two sides are fixed and the third side varies such that a triangle is still possible. Specifically, we are given two sides: \( a \) and \( b \), where \( a \leq b \). The third side must satisfy the triangle inequality.
Conditions Based on the Value of a and b
Case 1: \( a \leq \frac{b}{2} \)
Only one isosceles triangle is possible. The third side must be equal to \( b \) so that the triangle becomes: \( a, b, b \).
Case 2: \( \frac{b}{2} < a < b \)
In this case, two isosceles triangles can be formed:
- \( a, b, b \)
- \( a, a, b \)
Possible triangles: \( 4, 4, 5 \) and \( 4, 5, 5 \)
Case 3: \( a = b \)
In this scenario, no isosceles triangle with exactly two sides equal is possible. That’s because all three sides would become equal in any case, leading to an equilateral triangle, which is not counted here.
Total number of valid triangles (not necessarily isosceles): \( 2a - 1 \)
Summary Table
| Condition | Number of Isosceles Triangles | Example |
|---|---|---|
| \( a \leq \frac{b}{2} \) | 1 | \( a = 2, b = 5 \Rightarrow (2,5,5) \) |
| \( \frac{b}{2} < a < b \) | 2 | \( a = 4, b = 5 \Rightarrow (4,4,5), (4,5,5) \) |
| \( a = b \) | 0 (Only equilateral possible) | \( a = b = 5 \Rightarrow (5,5,5) \) |
Conclusion
This shortcut is extremely useful in number-based triangle formation problems, especially in geometry-based Logical Reasoning or CAT-style MCQs. Remember, the key is to check the ratio of the smaller side to half of the bigger side.
