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Triangles
Heron's formula, the law of cosines and classifying triangles.
Three sides determine a triangle completely (when they satisfy the triangle inequality). Heron's formula gives the area from the sides alone; the law of cosines gives each angle. The figure is drawn to scale from the numbers you typed.
Exemplu lucrat: triangle 3 4 5
Pas cu pas
- a = 3,\ b = 4,\ c = 5
Three sides (SSS). Check the triangle inequality: each side is less than the sum of the other two. ✓
- P = a + b + c = 12
Perimeter.
- s = \tfrac{P}{2} = 6,\quad A = \sqrt{s(s-a)(s-b)(s-c)} = 6
Heron's formula for the area.
- \cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{4}{5} \Rightarrow A \approx 36.870^\circ
Law of cosines for angle A (opposite side a).
- \cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{3}{5} \Rightarrow B \approx 53.130^\circ
Law of cosines for angle B (opposite side b).
- \cos C = \frac{a^2 + b^2 - c^2}{2ab} = 0 \Rightarrow C \approx 90.000^\circ
Law of cosines for angle C (opposite side c).
- A + B + C = 180.0^\circ
The angles add to 180° — a right, scalene triangle.
Revelează răspunsul
A = 6,\quad P = 12,\quad \angle \approx 36.9^\circ, 53.1^\circ, 90.0^\circ