In parallelogram ABCD, diagonal BD is perpendicular to side AD. Find AC if AD = 6 cm, BD = 5 cm.

Let a parallelogram ABCD be given, in which the diagonal BD is perpendicular to the side AD, side AD = 6 cm, BD = 5 cm, the diagonal AC = x cm, then:

AB² = AD² + BD²; AB² = 6² + 5² = 36 + 25 = 61 (cm²) – according to the Pythagorean theorem for a right-angled triangle ABD, since the side BD is perpendicular to the side AD and ∠ВDA = 90 °;

2 · AB² + 2 · AD² = BD² + AC² – since the sum of the squares of the diagonals of the parallelogram is equal to the sum of the squares of all its sides.

Substituting the values of the quantities into the formula, we get the equation:

2 · 61 + 2 · 6² = 5² + x²;

x = 13 (cm) – the length of the AC diagonal.

Answer: The length of the AC diagonal is 13 cm.