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Geometry · Triangles

Triangle 7 8 9

7,\ 8,\ 9

Step by step

  1. a = 7,\ b = 8,\ c = 9

    Three sides (SSS). Check the triangle inequality: each side is less than the sum of the other two. ✓

  2. P = a + b + c = 24

    Perimeter.

  3. s = \tfrac{P}{2} = 12,\quad A = \sqrt{s(s-a)(s-b)(s-c)} = 12 \sqrt{5} \approx 26.833

    Heron's formula for the area.

  4. \cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{2}{3} \Rightarrow A \approx 48.190^\circ

    Law of cosines for angle A (opposite side a).

  5. \cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{11}{21} \Rightarrow B \approx 58.412^\circ

    Law of cosines for angle B (opposite side b).

  6. \cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{2}{7} \Rightarrow C \approx 73.398^\circ

    Law of cosines for angle C (opposite side c).

  7. A + B + C = 180.0^\circ

    The angles add to 180° — a acute, scalene triangle.

Reveal the answer
A = 12 \sqrt{5} \approx 26.833,\quad P = 24,\quad \angle \approx 48.2^\circ, 58.4^\circ, 73.4^\circ