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Consider triangle \(\triangle ABC\) in the picture below, with \(\angle A = 30^{\circ}, \angle C= 45^{\circ}\) and \(AB=10\):

 

 

 

What are \(AC\) and \(BC\), in that order?

 Apr 11, 2022
 #1
avatar+2455 
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Draw altitude \(BM\)

 

\(BM\) splits \(\triangle ABC\) into 2 triangles: \(30-60-90\) and a \(45 - 45 - 90\) triangle. 

 

10 is the hypotenuse of the \(30 - 60 - 90\) triangle, meaning \(BM = 5\), and \(AM = 5\sqrt3\)

 

\(\triangle BCM \) has the angles \(45 - 45- 90\) , so \(CM = 5\), and \(\color{brown}\boxed{BC = 5\sqrt2}\)

 

We also know \(AC = AM + MC\). Subbing in the known values, we find that \(\color{brown}\boxed{AC = 5 \sqrt3 + 5}\)

 Apr 11, 2022
 #2
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+1

....or you could just use the sin law

 

10 / sin 45    =BC / sin 30       solve for BC

 

10 / sin 45  =  AC / sin 105     solve for AC

 Apr 11, 2022

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