Sss triangle

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

Correct result:

S =  42.789 cm2
h1 =  10.697 cm
h2 =  7.78 cm
h3 =  7.132 cm

Solution:

a=8 cm b=11 cm c=12 cm  s=(a+b+c)/2=(8+11+12)/2=312=15.5 cm S=s⋅ (s−a)⋅ (s−b)⋅ (s−c)=15.5⋅ (15.5−8)⋅ (15.5−11)⋅ (15.5−12)=42.789 cm2a=8 \ \text{cm} \ \\ b=11 \ \text{cm} \ \\ c=12 \ \text{cm} \ \\ \ \\ s=(a+b+c)/2=(8+11+12)/2=\dfrac{ 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) }=42.789 \ \text{cm}^2
S=a⋅ h12 h1=2⋅ S/a=2⋅ 42.7895/8=10.697 cmS=\dfrac{ a \cdot \ h_{1} }{ 2 } \ \\ h_{1}=2 \cdot \ S / a=2 \cdot \ 42.7895 / 8=10.697 \ \text{cm}
h2=2⋅ S/b=2⋅ 42.7895/11=7.78 cmh_{2}=2 \cdot \ S / b=2 \cdot \ 42.7895 / 11=7.78 \ \text{cm}
h3=2⋅ S/c=2⋅ 42.7895/12=7.132 cmh_{3}=2 \cdot \ S / c=2 \cdot \ 42.7895 / 12=7.132 \ \text{cm}

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