GATE 2025 EC – Question 7
A stick of length one meter is broken at two locations at distances of $b_1$ and $b_2$ from the origin (0), as shown in the figure. Note that $0<b_1<b_2<1$. Which one of the following is NOT a necessary condition for forming a triangle using the three pieces?
Note: All lengths are in meter. The figure shown is representative.
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Correct answer: (D) $b_1+b_2<1$
Explanation
The pieces have lengths $b_1$, $b_2-b_1$ and $1-b_2$ and sum to 1. A triangle exists only if every piece is shorter than 0.5, i.e. $b_1<0.5$, $b_2-b_1<0.5$ and $b_2>0.5$. These are options A, C and B. Option D is not required: for $b_1=0.4$, $b_2=0.7$ the pieces 0.4, 0.3, 0.3 form a triangle although $b_1+b_2=1.1>1$.