Sss triangle

Calculate the area and heights in the triangle ABC by sides a = 8cm, b = 11cm, c = 12cm

Correct answer:

S =  42.7895 cm²
h₁ =  10.6974 cm
h₂ =  7.7799 cm
h₃ =  7.1316 cm

Step-by-step explanation:

$$\begin{gathered} a=8\text{ cm } \\ b=11\text{ cm } \\ c=12\text{ cm } \\ s=(a+b+c)\mathrm{/}2=(8+11+12)\mathrm{/}2={\displaystyle \frac{31}{2}}=15.5\text{ cm } \\ S=\sqrt{s\cdot (s-a)\cdot (s-b)\cdot (s-c)}=\sqrt{15.5\cdot (15.5-8)\cdot (15.5-11)\cdot (15.5-12)}=42.7895{\text{cm}}^2 \end{gathered}$$

$$\begin{gathered} S={\displaystyle \frac{a\cdot h_1}{2}} \\ {h}_1=2\cdot S\mathrm{/}a=2\cdot 42.7895\mathrm{/}8=10.6974\text{ cm} \end{gathered}$$

$h_2=2\cdot S\mathrm{/}b=2\cdot 42.7895\mathrm{/}11=7.7799\text{ cm}$

$h_3=2\cdot S\mathrm{/}c=2\cdot 42.7895\mathrm{/}12=7.1316\text{ cm}$

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