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# Triangle geo

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Two angles of a triangle measure 30 and 45 degrees. If the side of the triangle opposite the 30-degree angle measures  $$12$$ units, what is the area of the triangle? Express your answer as a decimal to the nearest tenth.

Apr 15, 2022

#1
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Draw an altitude that forms both a 45-45-90 triangle and a 30-60-90 triangle

Because we have a 30-60-90 triangle, the base of the triangle is $${12 \over \sqrt3} = {4 \sqrt3}$$

We also have a 45-45-90 triangle. We know 1 of the legs is 12, meaning the other leg must also be 12.

Thus, the area of the triangle is $$12 \times (12 +4 \sqrt3) \div 2 \approx \color{brown}\boxed{113.6}$$

Here is a diagram: Apr 15, 2022
edited by BuilderBoi  Apr 15, 2022
edited by BuilderBoi  Apr 15, 2022
#2
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Another way

The third angle  =   105°

Using the Law of Sines  the side opposite the 45°  angle = x =  can be found as

x / sin 45 =  12 /sin 30

x =    12 sin 45 / sin 30  =    12 sqrt (2)

So....using the area of  a triangle knowing two sides  and  an included angle we have

A =  (1/2)  (12 sqrt 2 ) * ( 12 )  * [  sin (105°) ] =

Note :     {sin 105 = sin 75 }

72sqrt (2) *   sin (30 + 45)  =

72 sqrt (2)  [  sin 30 * cos 45  + sin 45  * cos 30 ] =

72 sqrt (2) [ sqrt 2 / 4  +  sqrt 6 / 4 ]  =

72 (2/4)  + 18 sqrt (12)  =

36 + 18 * 2 sqrt (3)   =

36 + 36 sqrt (3) =  36 (1 + sqrt (3) )  units^2 ≈   98.4  units^2   Apr 15, 2022
edited by CPhill  Apr 15, 2022