Triangle's centroid

In the triangle ABC the given lengths of its medians tc = 9, ta = 6. Let T be the intersection of the medians (triangle's centroid), and the point S is the center of the side BC. The magnitude of the CTS angle is 60°.
Calculate the length of the BC side to 2 decimal places.

Final Answer:

a =  10.58 cm

Step-by-step explanation:

$$\begin{gathered} tc=9\text{cm} \\ ta=6\text{cm} \\ CTS=60°\to \text{rad}=60°\cdot \frac{\pi }{180}=60°\cdot \frac{3.1415926}{180}=1.0472=\pi /3 \\ CT=\frac{2}{1+2}\cdot tc=\frac{2}{1+2}\cdot 9=6\text{cm} \\ ST=\frac{1}{1+2}\cdot ta=\frac{1}{1+2}\cdot 6=2\text{cm} \\ {x}^2=C{T}^2+S{T}^2-2\cdot CT\cdot ST\cdot \cos CTS \\ x=\sqrt{C{T}^2+S{T}^2-2\cdot CT\cdot ST\cdot \cos (CTS)}=\sqrt{6^2+2^2-2\cdot 6\cdot 2\cdot \cos 1.0472}=2\sqrt{7}\text{cm}\doteq 5.2915\text{cm} \\ a=2\cdot x=2\cdot 5.2915=4\sqrt{7}=10.58\text{cm} \end{gathered}$$

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