Calculate triangle

In the triangle ABC, calculate the sizes of all heights, angles, perimeters and its area, if given a-40cm, b-57cm, c-59cm

Correct answer:

o =  156 cm
S =  1087.4907 cm²
v₁ =  54.3745 cm
v₂ =  38.1576 cm
v₃ =  36.8641 cm
A =  40.2961 °
B =  67.1615 °
C =  72.5424 °

Step-by-step explanation:

$$\begin{gathered} a=40\text{ cm } \\ b=57\text{ cm } \\ c=59\text{ cm } \\ o=a+b+c=40+57+59=156\text{ cm} \end{gathered}$$

$$\begin{gathered} s=o\mathrm{/}2=156\mathrm{/}2=78\text{ cm } \\ S=\sqrt{s\cdot (s-a)\cdot (s-b)\cdot (s-c)}=\sqrt{78\cdot (78-40)\cdot (78-57)\cdot (78-59)}=114\sqrt{91}=1087.4907{\text{cm}}^2 \end{gathered}$$

$$\begin{gathered} S={\displaystyle \frac{a\cdot v_1}{2}} \\ {v}_1=2\cdot S\mathrm{/}a=2\cdot 1087.4907\mathrm{/}40=54.3745\text{ cm} \end{gathered}$$

$v_2=2\cdot S\mathrm{/}b=2\cdot 1087.4907\mathrm{/}57=4\sqrt{91}=38.1576\text{ cm}$

$v_3=2\cdot S\mathrm{/}c=2\cdot 1087.4907\mathrm{/}59=36.8641\text{ cm}$

$$\begin{gathered} \sin A=v_3:b=36.8641:57\doteq 0.6467 \\ A={\displaystyle \frac{180^{\circ }}{\pi }}\cdot \arcsin (v_3\mathrm{/}b)={\displaystyle \frac{180^{\circ }}{\pi }}\cdot \arcsin (36.8641\mathrm{/}57)=40.2961^{\circ }=40^{\circ }17^{\mathrm{\prime }}46\mathrm{"} \end{gathered}$$

$$\begin{gathered} \sin B=v_3:a=36.8641:40\doteq 0.9216 \\ B={\displaystyle \frac{180^{\circ }}{\pi }}\cdot \arcsin (v_3\mathrm{/}a)={\displaystyle \frac{180^{\circ }}{\pi }}\cdot \arcsin (36.8641\mathrm{/}40)=67.1615^{\circ }=67^{\circ }{9}^{\mathrm{\prime }}41\mathrm{"} \end{gathered}$$

$C=180-A-B=180-40.2961-67.1615=72.5424^{\circ }=72^{\circ }32^{\mathrm{\prime }}33\mathrm{"}$

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