In a parallelogram with sides of 8 cm and 12 cm and a cosine of 1/4, find the length of the dioganal.

From the triangle ABD, by the cosine theorem, we determine the length of the diagonal BD.

BD ^ 2 = AB ^ 2 + AD ^ 2 – 2 * AB * AD * CosA.

BD ^ 2 = 64 + 144 – 2 * 8 * 12 * 1/4 = 208 – 48 = 160.

ВD = √160 = 4 * √10 cm.

Let the angle BAD = X0, then the angle ABC = (180 – X) 0.

Cos (180 – X) = – CosX.

Then CosABS = -1 / 4.

In the triangle ABC, by the cosine theorem, we define the length of the AC.

AC ^ 2 = AB ^ 2 + BC ^ 2 – 2 * AB * BC * CosA.

AC ^ 2 = 64 + 144 – 2 * 8 * 12 * (-1 / 4) = 208 + 48 = 256.

AC = 16 cm.

Answer: The diagonals of the parallelogram are 16 cm and 4 * √10 cm.