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a, b, c are the lengths of three sides of a triangle ABC. If a, b, c are related by the relation a² + b² + c² = ab + bc + ca, then the value of sin²A + sin²B + sin²C is
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2 4 -
3√3 2 -
2 2 -
9 4
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Correct Option: D
a² + b² + c² = ab + bc + ca
⇒ 2a² + 2b² + 2c² = 2ab + 2bc + 2ca
⇒ a² + b² + b² + c² + c² + a² – 2ab – 2bc – 2ca = 0
⇒ a² + b² – 2ab + b² + c² – 2bc + c² + a² – 2ca = 0
⇒ (a – b)² + (b – c)² + (c – a)² = 0
⇒ a – b = 0
⇒ a = b
b – c = 0
⇒ b = c
c – a = 0
⇒ c = a
∴ ∆ ABC is an equilateral triangle.
∴ ∠A = ∠B = ∠ C = 60°
∴ sin²A + sin²B + sin²C = 3 sin²A = 3 × sin² 60°
| =3 × |  | √3 |  | ² | ||
| 2 |
| = | = | 4 | 4 |
