Question

Please solve q 95 ?

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Answer 1

The length of the longest side is `21`.

Explanation:

In a `DeltaABC`,

`rarrcosA=(b^2+c^2-a^2)/(2bc)`

`rarrArea=(1/2)a*bsinC`

Now, `Area` of `DeltaABD=(1/2)*9*8*sinx=36sinx`

`Area` of `DeltaADC=(1/2)*8*18*sinx=72sinx`

`Area` of `DeltaABC=(1/2)*9*18*sin2x=81sin2x`

`rarrDeltaABC=DeltaABD+DeltaADC`

`rarr81sin2x=36*sinx+72*sinx=108*sinx`

`rarr81*2cancel(sinx)*cosx=108*cancel(sinx)`

`rarrcosx=(108)/162=2/3`

Applying cosine law in `DeltaABC`, we get,

`rarrcos2x=(9^2+18^2-a^2)/(2*9*18)`

`rarr2cos^2x-1=(405-a^2)/324`

`rarr2*(2/3)^2-1=(405-a^2)/324`

`rarr2*(4/9)-1=(405-a^2)/324`

`rarr-36=405-a^2`

`rarra^2=405+36=441`

`rarra=21`

Also, note that

`rarrsin2x=2sinxcosx`

`rarrcos2x=2cos^2x-1`