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There's one triangle satisfying  \(\angle BCA = 60^{\circ}, AB = 12\) and \(AC = 10\) shown below:

 

 

The only possible value of \(BC\) is \(a\), as shown as above. What is \(a\)?

 Apr 15, 2022

Best Answer 

 #1
avatar+13595 
+2

What is a?

 

Hello Guest!

 

\(c^2=a^2+b^2-2bc\ cos \alpha\\ 12^2=a^2+10^2-2\cdot a\cdot 10\cdot cos\ 60°\\ a^2-10a-44=0\\ a=5\pm \sqrt{25+44}\\ \color{blue}a=13.31\)

laugh  !

 Apr 15, 2022
 #1
avatar+13595 
+2
Best Answer

What is a?

 

Hello Guest!

 

\(c^2=a^2+b^2-2bc\ cos \alpha\\ 12^2=a^2+10^2-2\cdot a\cdot 10\cdot cos\ 60°\\ a^2-10a-44=0\\ a=5\pm \sqrt{25+44}\\ \color{blue}a=13.31\)

laugh  !

asinus Apr 15, 2022
 #2
avatar+1384 
+1

Here's another way: Draw altitude \(AM\).

 

\(\triangle {AMC} \) is a 30-60-90 triangle, meaning \(CM = 5\) and \(AM = 5 \sqrt3\)

 

Now, \(\triangle {AMB}\) is a right triangle, so applying the Pythagorean Theorem to \(\triangle AMB\), we find that \(MB = \sqrt{69}\)

 

Since \(CB = CM + MB\)\(CB = \color{brown}\boxed{5 + \sqrt{69}}\)

 Apr 15, 2022

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