# a:c=12:7,b=3,α=2β find a,c and γ

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### 1 Answer

We will use the fact that ratios of sides o a triangle `a:b:c` are equal to ratios of angles `alpha:beta:gamma`. So with that in mind we have

`alpha=2beta => a=2b => a=6`

And since `a:c=12:7` we have

`6:c=12:7=>12c=42=>c=3.5`

Also by using the fact `a:b:c=alpha:beta:gamma` we have

`a:c=12:7=>alpha:gamma=12:7=>12gamma=7alpha`

Also we know that sum of all angles of triangle is equal to 180° so we have system of 3 equations:

`alpha=2beta`

`12gamma=7alpha`

`alpha+beta+gamma=180^o`

By solving this system of equations we get: `alpha=86^o24',` `beta=43^o12'` and `gamma=50^o24'`.

We could have calculated `gamma` by using law of cosines:

`a=b^2+c^2-2bc cos gamma`

Also we could have used law of sines to calculate `a` and `c`:

`a/sinalpha=b/sinbeta=c/singamma`