Question

The three sides of a triangle have lengths $a, b, c$. If $a^2 + b^2 + c^2 = ab + bc + ca$, then the triangle is:

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

Hint:

If sum of squared side differences is zero, all sides must be equal.

Answer 1

The Correct Option is A

Given: \[ a^2 + b^2 + c^2 = ab + bc + ca \] Rearrange: \[ a^2 + b^2 + c^2 - ab - bc - ca = 0 \] Multiply by 2: \[ (a-b)^2 + (b-c)^2 + (c-a)^2 = 0 \] Since each square is non-negative, all must be zero: \[ a = b = c \] Thus the triangle is equilateral. \[ \boxed{\text{Equilateral}} \]