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in Triangles by (23.5k points)
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If a, b, c are the sides of a triangle and a2 + b2 + c2 = bc + ca + ab, then the triangle is:

(a) equilateral 

(b) isosceles 

(c) right-angled 

(d) obtuse angled

1 Answer

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Best answer

(a) equilateral

a2 + b2 + c2 = ab + bc + ca 

⇒ a2 + b2 + c2 – ab – bc – ca = 0 

⇒ 2a2 + 2b2 + 2c2 – 2ab – 2bc – 2ca = 0 

⇒ (a – b)2 + (b – c)2 + (c – a)2 = 0 

Sum of perfect squares = 0 ⇒ Each term of the sum is zero 

⇒ (a – b) = 0 = (b – c) = (c – a) 

⇒ a = b = c 

⇒ The triangle is equilateral.

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