The sides of parallelogram ABCD are 2 cm and 6 cm. Calculate the diagonal AC if BD is √31 cm.

Let in the parallelogram ABCD AB = CD = 2 cm, and BC = AD = 6 cm.BD = √31 cm.

In triangle ABD we know all three sides. We calculate the cosine of the angle A by the cosine theorem:

BD² = AB² + AD² – 2 * AB * AD * cosA.

(√31) ² = 2² + 6² – 2 * 2 * 6 * cosA.

31 = 4 + 36 – 24cosA.

24cosA = 40 – 31.

24cosA = 9.

cosA = 9/24.

In a parallelogram, the sides adjacent to one side add up to 180 °, so the angle B is (180 ° – A).

Therefore, cosB = cos (180 ° – A) = -cosA.

In triangle ABC by the cosine theorem:

AC² = AB² + BC² – 2 * AB * BC * cosB.

AC² = 2² + 6² – 2 * 2 * 6 * (-cosA) = 4 + 36 + 24 * 9/24 = 40 + 9 = 49.

AC = √49 = 7 (cm).