Question : The length of the sides of a triangle is $a, b,$ and $c$ respectively if $a^{2} + b^{2} + c^{2} = a b + b c + c a$ , then the triangle is:
Option 1: Isosceles
Option 2: Equilateral
Option 3: Scalene
Option 4: Right-angled

Question

Question : If one angle of a triangle is equal to half the sum of the other two equal angles, then the triangle is:
Option 1: isosceles
Option 2: scalene
Option 3: equilateral
Option 4: right-angled

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Answer 1

Correct Answer: Equilateral

Solution : Given: $a^{2} + b^{2} + c^{2} = a b + b c + c a$
Multiplying both sides by 2,
$\Rightarrow 2 (a^{2} + b^{2} + c^{2}) = 2 (a b + b c + c a)$
$\Rightarrow a^{2} - 2 a b + b^{2} + b^{2} - 2 b c + c^{2} + c^{2} - 2 c a + a^{2} = 0$
$\Rightarrow (a - b)^{2} + (b - c)^{2} + (c - a)^{2} = 0$ .
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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Question : The length of the sides of a triangle is a,b, and c respectively if a2+b2+c2=ab+bc+ca, then the triangle is:Option 1: IsoscelesOption 2: EquilateralOption 3: ScaleneOption 4: Right-angled

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Question : The length of the sides of a triangle is $a, b,$ and $c$ respectively if $a^{2} + b^{2} + c^{2} = a b + b c + c a$ , then the triangle is:
Option 1: Isosceles
Option 2: Equilateral
Option 3: Scalene
Option 4: Right-angled