In rhombus ABCD, the height BK divides the side AD into segments AK = 12 cm and KD = 8 cm.

In rhombus ABCD, the height BK divides the side AD into segments AK = 12 cm and KD = 8 cm. Find the diagonal BD and the height BK.

Determine the length of the AD side. AD = AK + DK = 12 + 8 = 20 cm.

Since all sides of a rhombus are equal, then AB = BC = CD = AD = 20 cm.

Since BK is the height of the rhombus, the triangles ABK and BKD are rectangular.

Then in the right-angled triangle ABK, by the Pythagorean theorem, we define the leg BK.

BK ^ 2 = AB ^ 2 – AK ^ 2 = 400 – 144 = 256.

BK = 16 cm.

In a right-angled triangle ВDК, according to the Pythagorean theorem, we determine the length of the hypotenuse ВD.

BD ^ 2 = BK ^ 2 + DK ^ 2 = 16 ^ 2 + 8 ^ 2 = 256 + 64 = 320.

ВD = 8 * √5 cm.

Answer: The diagonal BD is 8 * √5, the height of the BC is 16 cm.