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 \(a\)?
+14913 What is a?
Hello Guest!
\(c^2=a^2+b^2-2bc\ cos \alpha\\ 12^2=a^2+10^2-2\cdot a\cdot 10\cdot cos\ 60°\\ a^2-10a-44=0\\ a=5\pm \sqrt{25+44}\\ \color{blue}a=13.31\)
!
+2666 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 \(CB = CM + MB\), \(CB = \color{brown}\boxed{5 + \sqrt{69}}\)
