The height of the rhombus is 24 cm, and its diagonals are 3: 4. Find the area of the rhombus.

Let the length of the diagonal BD = 3 * X cm, then AC = 4 * X cm.

At the rhombus, the diagonals intersect at right angles and are halved at the point of intersection.

Then AO = AC / 2 = 4 * X / 2 = 2 * X cm.

BO = BD / 2 = 3 * X / 2 cm.

In a right-angled triangle AOD, AD ^ 2 = AO ^ 2 + DO ^ 2 = 9 * X ^ 2/4 + 4 * X ^ 2 = (9 * X ^ 2 + 16 * X ^ 2) / 4 = 25 * X ^ 2/4.

AD = 5 * X / 2.

Let’s define the area of a rhombus in two ways.

Savsd = (AC * BD) / 2 = 4 * X * 3 * X / 2 = 6 * X2 cm2.

Savsd = AD * BH = 5 * X / 2 * 24 = 60 * X cm2.

Let’s equate both equations.

6 * X2 = 60 * X.

X = 60/6 = 10 cm.

Savsd = 60 * 10 = 600 cm2.

Answer: The area of the rhombus is 600 cm2.