Sides ratio

Calculate the circumference of a triangle with area 84 cm2 if a:b:c = 10:17:21

Result

o =  48 cm

Solution:

a1=10 b1=17 c1=21  S=84  s1=(a1+b1+c1)/2=(10+17+21)/2=24 S1=s1⋅ (s1−a1)⋅ (s1−b1)⋅ (s1−c1)=24⋅ (24−10)⋅ (24−17)⋅ (24−21)=84 k=S/S1=84/84=1 a=k⋅ a1=1⋅ 10=10 b=k⋅ b1=1⋅ 17=17 c=k⋅ c1=1⋅ 21=21 o=a+b+c=10+17+21=48=48  cm a_{ 1 } = 10 \ \\ b_{ 1 } = 17 \ \\ c_{ 1 } = 21 \ \\ \ \\ S = 84 \ \\ \ \\ s_{ 1 } = (a_{ 1 }+b_{ 1 }+c_{ 1 })/2 = (10+17+21)/2 = 24 \ \\ S_{ 1 } = \sqrt{ s_{ 1 } \cdot \ (s_{ 1 }-a_{ 1 }) \cdot \ (s_{ 1 }-b_{ 1 }) \cdot \ (s_{ 1 }-c_{ 1 }) } = \sqrt{ 24 \cdot \ (24-10) \cdot \ (24-17) \cdot \ (24-21) } = 84 \ \\ k = \sqrt{ S/S_{ 1 } } = \sqrt{ 84/84 } = 1 \ \\ a = k \cdot \ a_{ 1 } = 1 \cdot \ 10 = 10 \ \\ b = k \cdot \ b_{ 1 } = 1 \cdot \ 17 = 17 \ \\ c = k \cdot \ c_{ 1 } = 1 \cdot \ 21 = 21 \ \\ o = a+b+c = 10+17+21 = 48 = 48 \ \text{ cm }

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