Ta có:

a)

 Áp dụng định lí côsin tam giác ta có:

$b^{2} = a^{2} + c^{2} -2ac.cosB = 8^{2} + 5^{2} - 2.8.5.cos60 = 49$

$\Rightarrow b = 7$

$a^{2} = b^{2} + c^{2} - 2bc.cosA$

$\Rightarrow cosA = \frac{b^{2}+c^{2}-a^{2}}{2bc} = \frac{7^{2}+5^{2}-8^{2}}{2.5.7} = \frac{1}{7}$

$\Rightarrow \widehat{A} \approx 82^{o}$

$c^{2} = a^{2} + b^{2} -2ab.cosC$

$=> cosC = \frac{a^{2} + b^{2} - c^{2}}{2ab}$

$=> \widehat{C} = 38^{o}$

b) 

$S = \frac{1}{2}ac.sinB = \frac{1}{2}.8.5.sin60 = 10\sqrt{3}$

$S = \frac{1}{2}bh_{b}$

$h_{b} = \frac{2S}{b} = \frac{2.10\sqrt{3}}{7} = \frac{20\sqrt{3}}{7}$

c) 

$m_{a}^{2} = \frac{b^{2}+c^{2}}{2} - \frac{a^{2}}{4} = \frac{7^{2}+5^{2}}{2} - \frac{8^{2}}{4} = 21$

$=> m_{a} = \sqrt{21}$