The side of the rhombus refers to the larger diagonal as 5: 8. Find the height of the rhombus if the smaller diagonal is 30 cm.

By condition, BC / AC = 5/8.

Let the sides of the rhombus be 5 * X, then the large diagonal will be 8 * X.

Consider a right-angled triangle BOC, leg BO = BD / 2 = 30/2 = 15 cm, leg OC = AC / 2 = (8 * X) / 2 = 4 * X, hypotenuse BC = 5 * X.

Then, by the Pythagorean theorem, BC ^ 2 = BO ^ 2 + OC ^ 2.

(5 * X) ^ 2 = 15 ^ 2 + (4 * X) ^ 2.

25 * X ^ 2 – 16 * X ^ 2 = 225.

9 * X ^ 2 = 225.

X ^ 2 = 25.

X = 5.

Then BC = AB = AD = CD = 5 * 5 = 25 cm.

AC = 8 * 5 = 40 cm.

The area of ​​the rhombus is half the product of the diagonals.

S = АС * ВD / 2 = 40 * 30/2 = 600 cm2.

Also, the area of ​​the rhombus is equal to the product of the side of the rhombus by the height.

S = AD * BH = 25 * BH.

BH = 600/25 = 24 cm.

Answer: The height of the rhombus is 24 cm.