⁦Consequences of the cosine rule⁩

⁦In the Ukrainian Year 9 geometry book (section 2⁩, "⁦The cosine rule⁩") ⁦the sides of a triangle are a, b, c and the opposite angles are α, β, γ⁩. ⁦The cosine rule⁩: a^٢ = b^٢ + c^٢ − ٢bccosα. ⁦It gives ٢bccosα = b^٢ + c^٢ − a^٢⁩.

⁦A consequence of the cosine rule⁩. ⁦Let a be the length of the longest side of a triangle⁩. ⁦If a^٢ < b^٢ + c^٢, the triangle is acute⁩. ⁦If a^٢ > b^٢ + c^٢, it is obtuse⁩. ⁦If a^٢ = b^٢ + c^٢, it is right-angled⁩. ⁦The reason is the sign of the cosine of the largest angle⁩: ⁦positive, negative or zero⁩.

⁦Problem 1⁩. ⁦The sum of the squares of the diagonals of a parallelogram equals the sum of the squares of all its sides⁩: ⁦BD^٢ + AC^٢ = ٢a^٢ + ٢b^٢, where a and b are neighbouring sides⁩. ⁦It is proved by writing the cosine rule for triangles ABD and ACD (angles α and 180° − α) and adding⁩.

⁦Problem 2⁩. ⁦The cosine rule can give a quadratic equation⁩. ⁦A root that does not fit the condition (a length cannot be negative) is thrown out⁩. ⁦Solutions are written⁩: ⁦Given, Find, Solution, Answer⁩. ⁦The triangles are drawn to scale⁩: ⁦1 square on the board is 1 cm⁩.