a:c=12:7,b=3,α=2β find a,c and γ

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tiburtius | High School Teacher | (Level 2) Educator

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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


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


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




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`:


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