Consequences of the cosine rule
The type of a triangle from its longest side (acute, right or obtuse), the sum of the squares of the diagonals of a parallelogram and a problem about sides AB and BC. This follows the Ukrainian Year 9 geometry book, section 2 "The cosine rule".
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- Russia: grade 9
Guide
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: a2 = b2 + c2 − 2bccosα. It gives 2bccosα = b2 + c2 − a2.
A consequence of the cosine rule. Let a be the length of the longest side of a triangle. If a2 < b2 + c2, the triangle is acute. If a2 > b2 + c2, it is obtuse. If a2 = b2 + c2, 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: BD2 + AC2 = 2a2 + 2b2, 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.