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 point is S 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.

Correct answer:

a =  10.58 cm

Step-by-step explanation:

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

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