Question

A triangle has sides A, B, and C. If the angle between sides A and B is `(pi)/4`, the angle between sides B and C is `(7pi)/12`, and the length of B is 5, what is the area of the triangle?

Answer 1

Area`=17.075units^2`

Explanation:

`cancelpi^color(magenta)1/cancel4^color(magenta)1xxcancel180^color(magenta)45/cancelpi^color(magenta)1=45^@=anglec`

`(7cancelpi^color(magenta)1)/cancel12^color(magenta)1xxcancel180^color(magenta)15/cancelpi^color(magenta)1=105^@=anglea`

`180^@-(105^@+45^@)=30^@=angleb`

`A/(sinanglea)=B/(sinangleb)`

`A=(sin105^@xx5)/sin30^@`

`A=4.829629131/0.5`

`A=9.659258262`

`color(magenta)(A=9.659`units to the nearest 3 decimal places

`C/(sinanglec)=B/(sinangleb)`

`C=(5*sin45^@)/sin30^@`

`C=3.535533906/0.5`

`C=7.071067812`

`color(magenta)(C=7.071units` to the nearest 3 decimal places

Area`=1/2BCsinanglea`

`0.5*5*7.071067812*sin105^@`

Area`=17.07531755units^2`

Area`color(magenta)(=17.075units^2` to the nearest 3 decimal places

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Check:-

Hero's formula:-

`S=(a+b+c)/2`

`S=(9.659+5+7.071)/2`

Area`=sqrt(s(s-a)(s-b)(s-c))`

`S=10.865`units

Area`sqrt(10.865(10.865-9.659)(10.865-5)(10.865-7.071))`

`Area=sqrt(10.865(1.206)(5.865)(3.794))`

`Areacolor(magenta)(=17.075units^2` to the nearest 3 decimal places