In an isosceles triangle ABC, AB = BC = 6 cm, angle A = 30 °. Find the AC and the height BD of the triangle.

Let in an isosceles triangle ABC, AB = BC = 6 cm, angle A = 30 °, BD – the height of the triangle. Consider a right-angled triangle ABD, in which the leg BD lies opposite the angle A = 30 °, which means that it is equal to half the hypotenuse: BD = AB: 2 = 6: 2 = 3 (cm). The leg AD can be found using the Pythagorean theorem:
BD² + AD² = AB²;
AD² = AB² – BD²;
AD² = 6² – 3²;
AD² = 27;
АD ≈ 5.2 (cm), then АС = 2 ∙ АD = 2 ∙ 5.2 ≈ 10.4 (cm),
Answer: height BD = 3 cm, side AC ≈ 10.4 cm in triangle ABC.