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Question

# In the adjoining figure, if BC=a,AC=b,AB=c and ∠CAB=120∘, then which of the following is the correct relation?

A

a2 = b2 + c2 + 2bc

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B

a2 = b2 + c2 - 2bc

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C

a2 = b2 + c2 + bc

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D

a2 = b2 + c2 - bc

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Solution

## The correct option is C a2 = b2 + c2 + bc In △CDB, BC2=CD2+BD2 [Pythagoras theorem] BC2=CD2+(DA+AB)2 BC2=CD2+DA2+AB2+(2×DA×AB) ...(i) In △ADC, CD2+DA2=AC2 ...(ii) [Pythagoras Theorem] Here, ∠CAB=120∘ (given) ⇒∠CAD=60∘ (since ∠CAD and ∠CAB form a linear pair of angles) Also, cos60∘=ADAC AC=2AD ...(iii) Substituting the values from (ii) & (iii) in (i) we get, BC2=AC2+AB2+(AC×AB) a2=b2+c2+bc Alternatively, Since ∠A is an obtuse angle in △ABC so, BC2=AB2+AC2+2AB.AD =AB2+AC2+2×AB×12×AC [∵AD=ACcos60∘=12AC] =AB2+AC2+AB×AC ⇒a2=b2+c2+bc

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