SSC CGL Tier II 12 Apr 2015 Paper

Question : 40 of 100

Marks: +1, -0

a, b, c are the lengths of three sides of a triangle ABC. If a, b, c are related by the relation

a^{2}+b^{2}+c^{2}=a b+b c+c a

, then the value of

sin^2A + sin^2B + sin^2C

Solution:

(4)

a^{2}+b^{2}+c^{2}=a b+b c+c a

⇒ 2 a^{2}+2 b^{2}+2 c^{2}=2 a b+2 b c

+2 c a

⇒ a^{2}+b^{2}+b^{2}+c^{2}+c^{2}+a^{2}-

2 a b-2 b c-2 c a=0

⇒a^{2}+b^{2}-2 a b+b^{2}+c^{2}-2 b c

+c^{2}+a^{2}-2 c a=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

∴ ΔABC is an equilateral triangle.
∴ ∠A = ∠B = ∠ C = 60°

∴ \sin ^{2} A+\sin ^{2} B+\sin ^{2} C

=3 \sin ^{2} A=3 \times \sin ^{2} 60^{\circ}

=3 ×( {√{3}}/{2} )^{2}

= {3 ×3}/{4}=\ {9}/{4}

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