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.
Show Hint
For triangle formation, the sum of any two sides must be greater than the third side. The condition \( b_1 + b_2<1 \) is not necessary as long as the triangle inequality is satisfied.
Step 1: Apply the triangle inequality theorem. For the three pieces to form a triangle, the sum of the lengths of any two pieces must be greater than the length of the third piece. Step 2: Analyze the options. (A) \( b_1<0.5 \) is a necessary condition. If \( b_1 \) were greater than or equal to 0.5, the other pieces would be too small to form a triangle. (B) \( b_2>0.5 \) is necessary because, if \( b_2 \leq 0.5 \), the sum of the two smaller pieces would not be enough to form a triangle. (C) \( b_2<b_1 + 0.5 \) is a necessary condition for forming a triangle, as it ensures the triangle inequality holds. (D) \( b_1 + b_2<1 \) is NOT a necessary condition for forming a triangle. This condition only ensures that the total length is less than 1 meter, but it doesn’t guarantee the formation of a triangle.
