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Area of Triangles

 May 21, 2019
 #1
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27.

 

K  =  area of triangle  =  \(\frac12\) ( base )( height )

 

Let the base  =  a

and the height  =  c sin B

 

Let's get  c  in terms of what we know using the Law of Sines

 

\(\frac{c}{\sin C}\,=\,\frac{a}{\sin A}\\~\\ c\,=\,\frac{a\sin C}{\sin A}\)

 

So the height  =  c sin B  =  (\(\frac{a\sin C}{\sin A}\)) sin B  =  \(\frac{a\sin B\sin C}{\sin A}\)

 

K  =  area of triangle  =  \(\frac12\) ( base )( height )  =  \(\frac12\) ( a )( \(\frac{a\sin B\sin C}{\sin A}\) )   =   \(\frac{a^2\sin B\sin C}{2\sin A}\)

 

\(K\,=\,\frac{a^2\sin B\sin C}{2\sin A}\)

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 May 21, 2019

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