Question: Let a, b and c be the sides of a triangle ABC. Is the triangle equilateral?<br>Statement-I: a^{2}+b^{2}+c^{2}=(ab+bc+ca)<br>Statement-II: 3a^{2}+3b^{2}+4c^{2}=2ab+4bc+4ca

Consider the following for the next ten (10) items that follow : Each item contains a Question followed by two Statements. Answer each item using the following instructions : Choose option (a) If the Question can be answered by one of the Statements alone, but not by the other. (b) If the Question can be answered by either Statement alone. (c) If the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone. (d) If the Question cannot be answered even by using both Statements together.

  1. A. If the Question can be answered by one of the Statements alone, but not by the other.
  2. B. If the Question can be answered by either Statement alone. ✓
  3. C. If the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.
  4. D. If the Question cannot be answered even by using both Statements together.

Correct Answer: B. If the Question can be answered by either Statement alone.

Explanation

Statement-I translates to a^2 + b^2 + c^2 - ab - bc - ca = 0, which can be algebraically factored into \frac{1}{2}[(a-b)^2 + (b-c)^2 + (c-a)^2] = 0. This sum of squares equals zero only if a = b = c, proving the triangle is equilateral. Statement-II gives 3a^2 + 3b^2 + 4c^2 - 2ab - 4bc - 4ca = 0, which cleanly groups into the identity (a-b)^2 + 2(a-c)^2 + 2(b-c)^2 = 0. By the same sum of squares logic, this also implies a = b = c. Since both statements independently force the sides to be equal, the question can be answered by either statement alone.

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