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In a triangle A B C , if cos A=(sinB)/(2...

In a triangle `A B C` , if `cos A=(sinB)/(2sinC)` , show that the triangle is isosceles.

Text Solution

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By the sine rule, we have
`a/("sin A")=b/("sin B")=c/("sin C")="k (sky)"`
`rArr" "("sin A")/a=("sin B")/b=("sin C")/c="k (sky)"`
`rArr" "sinA=ka, sinB=kband sinC=kc`.
`:." "cosA=(sinB)/(2sinC)rArr((b^(2)+c^(2)-a^(2)))/(2bc)=(kb)/(2kc)`
`rArr" "(b^(2)+c^(2)-a^(2))=b^(2)`
`rArr" "c^(2)=a^(2)rArr|c|=|a|`
`rArr" "AB=BC`.
Hence, `DeltaABC`, is isosceles.
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