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For an acute-angled triangle Δ, under which condition can two distinct triangles be formed?

1. The side a is smaller than b sin A.
2. The side a is greater than b sin A but less than b.
3. The side a is smaller than b sin A and greater than b.
4. The side a is greater than b sin A and equal to b.

Correct answer: The side a is greater than b sin A but less than b.

Solution

For two distinct triangles to form in an acute-angled triangle, the side 'a' must be greater than b sin A (to ensure a triangle exists) but less than 'b' (to avoid forming a single triangle).

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