Question

If \( A, B \) and \( C \) are the measures of the angles of \( \triangle A B C \) and \( a, b \) and \( c \) are the lengths of the sides opposite to the angles \( A, B \) and \( C \), then prove that \( a^{2}=b^{2}+c^{2}-2 b c \cos A \).

Answer 1

Consider triangle ABC with sides a, b, and c opposite to angles A, B, and C respectively.

According to the Law of Cosines, we have the formula: a² = b² + c² − 2bc cos A.

This formula can be derived from the coordinates of the points in the triangle or by applying the Pythagorean theorem in the context of the triangle.

Thus, we conclude that the relation a² = b² + c² − 2bc cos A holds true.

Therefore, we have proved that a² = b² + c² − 2bc cos A.