Question : The length of the sides of a triangle is $a, b,$ and $c$ respectively if $a^2 + b^2 + c^ 2 = ab + bc + ca$, then the triangle is:
Option 1: Isosceles
Option 2: Equilateral
Option 3: Scalene
Option 4: Right-angled
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Correct Answer: Equilateral
Solution : Given: $a^2 + b^2 + c^2 = ab + bc + ca$ Multiplying both sides by 2, $⇒2(a^2 + b^2 + c^2) = 2(ab + bc + ca)$ $⇒a^2 - 2ab + b^2 + b^2 - 2bc + c^2 + c^2 - 2ca + a^2 = 0$ $⇒(a-b)^2 + (b-c)^2 + (c-a)^2 = 0$. Since the sum of squares is zero, each term must be zero, which implies that $a=b$, $b=c$, and $c=a$. Therefore, the triangle is equilateral because all its sides are equal. Hence, the correct answer is Equilateral.
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