Heron backlaw

Calculate the missing side in a triangle with sides 25 and 13 and area 152.

x₁ = 23.7516
x₂ = 31.9978

a = 25 b = 13 S = 152 s = (a + b + c) / 2 S = s (s - a) (s - b) (s - c) S^{2} = (a + b + c) / 2 \cdot ((a + b + c) / 2 - a) \cdot ((a + b + c) / 2 - b) \cdot ((a + b + c) / 2 - c) 16 S^{2} = (a + b + c) \cdot ((a + b + c) - 2 a) \cdot ((a + b + c) - 2 b) \cdot ((a + b + c) - 2 c) 16 S^{2} = (a + b + c) \cdot (b + c - a) \cdot (a - b + c) \cdot (a + b - c) 16 S^{2} = - a^{4} + 2 a^{2} b^{2} - b^{4} + 2 a^{2} c^{2} + 2 b^{2} c^{2} - c^{4} S = \frac{a \cdot h}{2} = \frac{a \cdot b \cdot sin α}{2} α = arcsin (\frac{2 \cdot S}{a \cdot b}) = arcsin (\frac{2 \cdot 1 5 2}{2 5 \cdot 1 3}) ≐ 1.2093 rad x_{1} = a^{2} + b^{2} - 2 \cdot a \cdot b \cdot cos α = 2 5^{2} + 1 3^{2} - 2 \cdot 25 \cdot 13 \cdot cos 1.2093 ≐ 23.7516 Verifying Solution: s_{1} = \frac{a + b + x _{1}}{2} = \frac{2 5 + 1 3 + 2 3 . 7 5 1 6}{2} ≐ 30.8758 S_{1} = s_{1} \cdot (s_{1} - a) \cdot (s_{1} - b) \cdot (s_{1} - x_{1}) = 30.8758 \cdot (30.8758 - 25) \cdot (30.8758 - 13) \cdot (30.8758 - 23.7516) = 152 S_{1} = S

sin α = sin (π - α) α_{2} = π - α = 3.1416 - 1.2093 ≐ 1.9322 rad x_{2} = a^{2} + b^{2} - 2 \cdot a \cdot b \cdot cos (α_{2}) = 2 5^{2} + 1 3^{2} - 2 \cdot 25 \cdot 13 \cdot cos 1.9322 ≐ 31.9978 s_{2} = \frac{a + b + x _{2}}{2} = \frac{2 5 + 1 3 + 3 1 . 9 9 7 8}{2} ≐ 34.9989 S_{2} = s_{2} \cdot (s_{2} - a) \cdot (s_{2} - b) \cdot (s_{2} - x_{2}) = 34.9989 \cdot (34.9989 - 25) \cdot (34.9989 - 13) \cdot (34.9989 - 31.9978) = 152 S_{2} = S

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