Triangle solving calculation

Solve the triangle ABC if the side a = 52 cm, the height on the other side is vb = 21 cm, and the triangle's area is S = 330 cm².

Final Answer:

b = 31.4286 cm
c₁ = 25.1992 cm
c₂ = -25.1992 cm

a = 52 cm v_{2} = 21 cm S = 330 cm^{2} S = \frac{b \cdot v _{2}}{2} b = 2 \cdot S / v_{2} = 2 \cdot 330 / 21 = \frac{2 2 0}{7} cm = 31.4286 cm

S = s \cdot (s - a) (s - b) (s - c) s = (a + b + c) / 2 = a / 2 + b / 2 + c / 2 S = (a / 2 + b / 2 + c / 2) \cdot (a / 2 + b / 2 + c / 2 - a) (a / 2 + b / 2 + c / 2 - b) (a / 2 + b / 2 + c / 2 - c) S = (a / 2 + b / 2 + c / 2) \cdot (b / 2 + c / 2 - a / 2) (a / 2 + c / 2 - b / 2) (a / 2 + b / 2 - c / 2) 330 \cdot 330 = (a / 2 + b / 2 + c / 2) \cdot (b / 2 + c / 2 - a / 2) \cdot (a / 2 + c / 2 - b / 2) \cdot (a / 2 + b / 2 - c / 2) 330 \cdot 330 = (52 / 2 + 31.428571428571 / 2 + c / 2) \cdot (31.428571428571 / 2 + c / 2 - 52 / 2) \cdot (52 / 2 + c / 2 - 31.428571428571 / 2) \cdot (52 / 2 + 31.428571428571 / 2 - c / 2) - 461.40688775515 c^{2} + 292993.534 = 0 461.40688775515 c^{2} - 292993.534 = 0 c_{1, 2} = \pm 292993.53436069 / 461.40688775515 = \pm 25.199213245 c_{1} = 25.199213245 c_{2} = - 25.199213245 c = 26.487 cm c = 25.199 cm s = (a + b + c) / 2 = (52 + 31.4286 + 25.199) / 2 ≐ 54.3138 cm S_{1} = s \cdot (s - a) \cdot (s - b) \cdot (s - c) = 54.3138 \cdot (54.3138 - 52) \cdot (54.3138 - 31.4286) \cdot (54.3138 - 25.199) ≐ 289.3683 cm^{2} c_{1} = S_{1} = S = 25.1992 cm

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