# A triangle has sides A, B, and C. Sides A and B are of lengths 9 and 12, respectively, and the angle between A and B is pi/3. What is the length of side C?

Apr 1, 2016

$C = 3 \sqrt{13}$

#### Explanation:

You use the Cosine rule for this

Note that $\cos \left(\frac{\pi}{3}\right) \to \cos \left({60}^{o}\right) = \frac{1}{2}$

For the notation of this question and my diagram.

Cosine rule$\to {C}^{2} = {A}^{2} + {B}^{2} - 2 A B \cos \left(a\right)$

$\implies {C}^{2} = {9}^{2} + {12}^{2} - 2 \left(9\right) \left(12\right) \left(\frac{1}{2}\right)$

${C}^{2} = 81 + 144 - 108 = 117$

$\implies {C}^{2} = \sqrt{{3}^{3} \times 13}$

$\implies C = 3 \sqrt{13}$

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Comment:

I always understood that the standard practice of labelling was that capital letters were used for the vertices (angles) and that small letters were used for the sides.