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# Consider triangle in the picture below, with and :

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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
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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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....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