Heron backlaw

Calculate the unknown side in a triangle with sides 24 and 40 and area 299.6.

Correct answer:

x₁ = 25.9985
x₂ = 60.6307

a = 24 b = 40 S = 299.6 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 2 9 9 . 6}{2 4 \cdot 4 0}) ≐ 0.6741 rad x_{1} = a^{2} + b^{2} - 2 \cdot a \cdot b \cdot cos α = 2 4^{2} + 4 0^{2} - 2 \cdot 24 \cdot 40 \cdot cos 0.6741 ≐ 25.9985 Verifying Solution: s_{1} = \frac{a + b + x _{1}}{2} = \frac{2 4 + 4 0 + 2 5 . 9 9 8 5}{2} ≐ 44.9992 S_{1} = s_{1} \cdot (s_{1} - a) \cdot (s_{1} - b) \cdot (s_{1} - x_{1}) = 44.9992 \cdot (44.9992 - 24) \cdot (44.9992 - 40) \cdot (44.9992 - 25.9985) = \frac{1 4 9 8}{5} = 299 \frac{3}{5} = 299.6 S_{1} = S

sin α = sin (π - α) α_{2} = π - α = 3.1416 - 0.6741 ≐ 2.4675 rad x_{2} = a^{2} + b^{2} - 2 \cdot a \cdot b \cdot cos (α_{2}) = 2 4^{2} + 4 0^{2} - 2 \cdot 24 \cdot 40 \cdot cos 2.4675 ≐ 60.6307 s_{2} = \frac{a + b + x _{2}}{2} = \frac{2 4 + 4 0 + 6 0 . 6 3 0 7}{2} ≐ 62.3153 S_{2} = s_{2} \cdot (s_{2} - a) \cdot (s_{2} - b) \cdot (s_{2} - x_{2}) = 62.3153 \cdot (62.3153 - 24) \cdot (62.3153 - 40) \cdot (62.3153 - 60.6307) = \frac{1 4 9 8}{5} = 299 \frac{3}{5} = 299.6 S_{2} = S

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