Question

If $a,b,c$ are the sides of a triangle and $a^2+b^2+c^2=bc+ca+ab$, then the triangle is :

• A. equilateral
• B. isosceles
• C. right-angled
• D. obtuse angled

Answer 1

The correct option is A

equilateral

Explanation for the correct answer:

Option $\left(A\right)$: equilateral

$a^2+b^2+c^2=ab+bc+ca$

$\Rightarrow a^2+b^2+c^2–ab–bc–ca=0$

$\Rightarrow 2a^2+2b^2+2c^2–2ab–2bc–2ca=0\text{(multiplying 2 both side)}$

$\Rightarrow {\left(a-b\right)}^2+{\left(b-c\right)}^2+{\left(c-a\right)}^2=0$

Sum of perfect squares $=0\Rightarrow$ Each term of the sum is zero

$\Rightarrow \left(a-b\right)=\left(b-c\right)=\left(c-a\right)=0$

$\Rightarrow a=b=c$
Thus, the triangle is equilateral.

Therefore, Option $\left(A\right)$ is the correct answer.