Finding the third side of a triangle given the area Perimeter = ( a

Michelle Mendoza 2022-07-15 Answered
Finding the third side of a triangle given the area
Perimeter = ( a + b + c ) 2
Area = A = p ( p a ) ( p b ) ( p c )
How do I simplify the above two equations to solve for c?
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Answers (2)

treccinair
Answered 2022-07-16 Author has 18 answers
By cosine rule,
(1) c 2 = a 2 + b 2 2 a b cos γ ,
And from the formula for the area
2 S = a b sin γ , sin γ = 2 S a b ,
so, with
cos γ = ± 1 sin 2 γ = ± 1 4 S 2 a 2 b 2 = ± a 2 b 2 4 S 2 a b
(1) becomes
c 2 = a 2 + b 2 ± 2 a 2 b 2 4 S 2 .
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Holetaug
Answered 2022-07-17 Author has 8 answers
s is the conventional way of representing the semiperimeter a + b + c 2 of the triangle with sides a , b , c
s ( s a ) ( s b ) ( s c ) = a 4 b 4 c 4 + 2 b 2 c 2 + 2 c 2 a 2 + 2 a 2 b 2 4 = Δ
Solving for c,
c = a 2 + b 2 ± 2 a 2 b 2 4 Δ 2
where area = Δ
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